Method, apparatus and medium for decoding or encoding

ABSTRACT

A method of decoding or encoding that includes generating a new multi-transform data structure by embedding one or more non-recursive transforms into a larger recursive transform. The method may further include receiving information regarding a target data block and determining whether to use a recursive transform or a non-recursive transform. When the determination is to use the recursive transform, the method may include generating the recursive transform using a multi-transform data structure and causing the target data block to be encoded or decoded using the generated recursive transform. If not, the method may include causing the target data block to be encoded or decoded using one of the one or more the non-recursive transforms embedded in the multi-transform data structure.

PRIORITY INFORMATION

This is a continuation application of U.S. application Ser. No.16/235,336 filed on Dec. 28, 2018, which claims priority from U.S.Provisional Application No. 62/670,704, filed on May 11, 2018, and U.S.Provisional Application No. 62/668,065, filed May 7, 2018, each filed inthe United States Patent & Trademark Office, the disclosures of whichare incorporated herein by reference in their entirety.

BACKGROUND Field

This disclosure is directed towards video compression technologies. Inparticular, the present disclosure is directed towards next-generationvideo coding technologies including video coding/decoding technologiesbeyond High Efficiency Video Coding (HEVC), such as Versatile VideoCoding (VVC). VVC is a video coding standard that is currently underdevelopment. The disclosed subject matter may be used in the context ofVVC. More specifically, an aspect of the disclosure is directed towardsa method, device and computer medium for encoding or decoding based onembedding one orthonormal transform into a larger orthonormal transform.

Description of Related Art

Video coding and decoding using inter-picture prediction with motioncompensation has been known for decades. Uncompressed digital video canconsist of a series of pictures, each picture having a spatial dimensionof, for example, 1920×1080 luminance samples and associated chrominancesamples. The series of pictures can have a fixed or variable picturerate (informally also known as frame rate), of, for example 60 picturesper second or 60 hertz (Hz). Uncompressed video has significant bitraterequirements. For example, 1080p60 4:2:0 video at 8 bit per sample(1920×1080 luminance sample resolution at 60 Hz frame rate) requiresclose to 1.5 Gbit/s bandwidth. An hour of such video requires more than600 GByte of storage space.

One purpose of video coding and decoding can be the reduction ofredundancy in the input video signal, through compression. Compressioncan help reduce aforementioned bandwidth or storage space requirements,in some cases by two orders of magnitude or more. Both lossless andlossy compression, as well as a combination thereof can be employed.Lossless compression refers to techniques where an exact copy of theoriginal signal can be reconstructed from the compressed originalsignal. When using lossy compression, the reconstructed signal may notbe identical to the original signal, but the distortion between originaland reconstructed signal is small enough to make the reconstructedsignal useful for the intended application. In the case of video, lossycompression is widely employed. The amount of distortion tolerateddepends on the application; for example, users of certain consumerstreaming applications may tolerate higher distortion than users oftelevision contribution applications. The compression ratio achievablecan reflect that: higher allowable/tolerable distortion can yield highercompression ratios.

A person of ordinary skill generally understands videocompression/decompression technology. In general, to compress video orimage data, a series of functional steps are performed to result in acompressed video or image data block. Although an image, such as a 360°image (e.g., captured by a 360° camera) may be suitable for compression,for ease of reading, compression of a video data block will beexplained. To generate a compressed video data block, under conventionalstandards (e.g., H.264, H.265) an uncompressed video sample stream thatis received from a video source may be partitioned or parsed, whichresults in a block of samples of two or more reference pictures.

Bi-Prediction can relate to techniques where a prediction unit (PU),such as a predicted block of samples, can be predicted from two motioncompensated blocks of samples of two or more reference pictures.Bi-prediction was first introduced into video coding standards inMPEG-1, and has also been included in other video coding technologiesand standards such as MPEG-2 Part 2 (or H.262), H.264 and H.265 as well.

When decompressing a compressed video file, during the reconstruction ofa sample of a bi-predicted PU, motion compensated and interpolated inputsamples from each reference block can be multiplied by a weightingfactor that can be different for each reference block, and such weightedsample values of the two reference blocks can be added to generate thesample under reconstruction. Such sample can be processed further bymechanisms such as loop filtering.

In MPEG-1 and MPEG-2, the weighting factors can be determined based onthe relative temporal distance between the picture to which the PU underreconstruction belongs to, and the two reference pictures. This ispossible because, in MPEG-1 and MPEG-2, one of the two reference I or Ppictures was in the “past”, and the other in the “future” (in terms ofpresentation order) of the B-picture under reconstruction, and becausein MPEG-1 and MPEG-2, there was a well-defined timing relationshipestablished for any picture under reconstruction in relation to itsreference pictures.

Starting with H.264, the reference picture selection concepts forbi-predicted pictures were relaxed such that the reference pictures onlyneeded to be earlier in decoding order, but not in presentation order.Further, the notion of time was also relaxed in that neither H.264 norH.265 requires a constrained/fixed picture interval in the time domain.Therefore, a decoder cannot calculate weighting factors any more basedon the timing information available in the bitstream. Instead, H.264 andH.265 include a “default” of 0.5 as the weighting factor for thereference samples of a bi-predicted picture. This default can beoverwritten by syntax available in the slice header known aspred_weight_table( ). The default of 0.5 or the information in thepred_weight_table may apply to all bi-predicted PUs in a given slice.

Non-Patent Literature 1 shows the H.265/HEVC standard. However, a needfor standardization of future video coding technology with a compressioncapability that significantly exceeds that of the HEVC standard(including its extensions) has been studied by the inventors.

Non-Patent Literature 2 discloses a recently launched standardizationformat for next-generation video coding beyond HEVC called VersatileVideo Coding (VVC) with a version VTM (VVC Test Model). VVC maygenerally provide a large (e.g., 64-point or higher) transform coreusing a 10-bit integer matrix.

Problem(s) to be Solved

Different from DCT-2, of which fast methods have been extensivelystudied, the implementation of DST-7 is still much less efficient thanDCT-2, e.g., matrix multiplication based in VTM 1 of Non-PatentLiterature 2. In Non-Patent Literature 3, a method is proposed toapproximate different types of DCTs and DSTs in JEM7 by applyingadjustment stages to a transform in the DCT-2 family, which includesDCT-2, DCT-3, DST-2 and DST-3, and the adjustment stage refers to amatrix multiplication using a sparse matrix which requires relativelyless operation counts. In Non-Patent Literature 4, a method forimplementing N-point DST-7 using 2N+1 point Discrete Fourier Transform(DFT) is proposed.

However, additional transform types in Adaptive Multiple Transforms(AMT) increase the complexity and implementation cost(s), sinceadditional transform cores need to be stored in on-chip memory forpractical implementation, and transform cores need to be loaded whenswitching the transform type from one block to another. In HEVC, asmaller DCT-2 transform is a part of the larger DCT-2 transform, whichsupports reusing the same logic for implementing different sizes oftransforms. However, with AMT, such a feature is missing, other thanDCT-2, a smaller transform cannot be re-used for a larger transform,which means different logic may need to be designed for implementingdifferent sizes of transforms.

The DCT-2 in HEVC is implemented using partial butterfly, which supportsidentical outputs between matrix multiplication and partial butterfly,i.e., dual implementation. However, half-size matrix multiplication isstill needed, which is still a computation burden, for example, partialbutterfly of 32-point HEVC DCT-2 requires 2×2, 4×4, 8×8 and 16×16 matrixmultiplications.

Listing of Related Art

Non-Patent Literature 1: “Transform design for HEVC with 16 bitintermediate data representation”; published by: Joint CollaborativeTeam on Video Coding (JCT-VC) of ITU-T SG16 WP3 and ISO/IECJTC1/SC29/WG11, 5th Meeting: Geneva, CH, Mar. 16-23, 2011 (Document:JCTVC-E243). The H.265/HEVC (High Efficiency Video Coding) standard mayinclude Version 1 (2013), Version 2 (2014), Version 3 (2015) and Version4 (2016) published by ITU-T VCEG (Q6/16) ISO/IEC MPEG (JTC 1/SC 29/WG11).

Non-Patent Literature 2: “Algorithm description for Versatile VideoCoding and Test Model 1 (VTM 1)”; published by Joint Video Experts Team(JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, 10th Meeting:San Diego, US, April 10-20, 2018 (Document: JVET-J1002-v2).

Non-Patent Literature 3: A. Said, H. Egilmez, V. Seregin, M. Karczewicz,“Complexity Reduction for Adaptive Multiple Transforms (AMTs) usingAdjustment Stages”, Joint Video Exploration Team (JVET) of ITU-T SG 16WP 3 and ISO/IEC JTC 1/SC 29/WG 11, San Diego, US, 10-20 Apr. 2018(Document: JVET-J0066).

Non-Patent Literature 4: M. Koo, J. Heo, J. Nam, N. Park, J. Lee, J.Choi, S. Yoo, H. Jang, L. Li, J. Lim, S. Paluri, M. Salehifar, S. Kim,“Description of SDR video coding technology proposal by LG Electronics,”Joint Video Exploration Team (JVET) of ITU-T SG 16 WP 3 and ISO/IEC JTC1/SC 29/WG 11, San Diego, US, 10-20 Apr. 2018 (Document: JVET-J0017).

SUMMARY

According to an aspect of the disclosure, a method of decoding orencoding may include generating a new multi-transform data structure byembedding one or more non-recursive transforms into a larger recursivetransform. The method may further include receiving informationregarding a target data block for encoding or decoding, the target datablock for encoding or decoding being one of: a compressed video or imagedata block or an uncompressed video or image data block. The method ofdecoding or encoding may further include determining whether to use arecursive transform or a non-recursive transform for the encoding ordecoding of the target data block. The method of decoding or encodingmay further include, when a result of the determination is to use therecursive transform: generate the recursive transform using themulti-transform data structure and cause or transmit information thatcauses the target data block to be encoded or decoded using thegenerated recursive transform. The method of encoding or decoding mayfurther include cause or transmit information that causes the targetdata block to be encoded or decoded using one of the one or morenon-recursive transforms embedded in the multi-transform data structure.

The method of encoding or decoding may further include that at leastpart of the one of the one or more non-recursive transforms of themulti-transform data structure is embedded in at least part of one halfof the larger recursive transform of the multi-transform data structure.

According to an aspect of the disclosure, an apparatus may include atleast one memory configured to store computer program code and at leastone processor configured to access the at least one memory and operateaccording to the computer program code. The computer program code mayinclude first applying code, second applying code, third applying code,fourth applying code, fifth applying code, and sixth applying code, butis not limited in the number of code segments that may be included.

The first applying code may be configured to generate a newmulti-transform data structure by embedding one or more non-recursivetransforms into a larger recursive transform. The second applying codemay be configured to cause the at least one processor to receiveinformation regarding a target data block for encoding or decoding, thetarget data block for encoding or decoding being one of: a compressedvideo or image data block or an uncompressed video or image data block.The third applying code may be configured to cause the at least oneprocessor to determine, for the encoding or decoding of the target datablock, whether to use a recursive transform or a non-recursive transformfor the encoding or decoding of the target data block.

The fourth applying code may be configured to cause the at least oneprocessor to when a result of the determination is to use the recursivetransform: generate the recursive transform using the multi-transformdata structure and cause or transmit information that causes the targetdata block to be encoded or decoded using the generated recursivetransform. The fifth applying code may be configured to cause the atleast one processor to when a result of the determination is to use thenon-recursive transform of the multi-transform data structure: cause ortransmit information that causes the target data block to be encoded ordecoded one of the one or more non-recursive transforms embedded in themulti-transform data structure.

The apparatus for encoding or decoding may further include that at leastpart of the one of the one or more non-recursive transforms of themulti-transform data structure is embedded in at least part of one halfof the larger recursive transform of the multi-transform data structure.

According to an aspect of the disclosure, a non-transitorycomputer-readable storage medium may store instructions that cause oneor more processors to execute a number of steps. The number of steps mayinclude generating a new multi-transform data structure by embedding oneor more non-recursive transforms into a larger recursive transform. Thenumber of steps may further include receiving information regarding atarget data block for encoding or decoding, the target data block forencoding or decoding may be one of: a compressed video or image datablock or an uncompressed video or image data block. The number of stepsmay further include determining, for the encoding or decoding of thetarget data block, whether to use a recursive transform or anon-recursive transform for the encoding or decoding of the target datablock.

The number of steps may further include, when a result of thedetermination is to use the recursive transform: generate the recursivetransform using the multi-transform data structure and cause or transmitinformation that causes the target data block to be encoded or decodedusing the generated recursive transform. The number of steps may furtherinclude, when a result of the determination is to use the non-recursivetransform of the multi-transform data structure: causing or transmittinginformation that causes the target data block to be encoded or decodedusing one of the one or more non-recursive transforms embedded in themulti-transform data structure, wherein at least part of the one of theone or more non-recursive transforms of the multi-transform datastructure is embedded in at least part of one half of the largerrecursive transform of the multi-transform data structure.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features, the nature, and various advantages of the disclosedsubject matter will be more apparent from the following detaileddescription and the accompanying drawings in which:

FIG. 1 is a schematic illustration of a simplified block diagram of acommunication system in accordance with an embodiment.

FIG. 2 is a schematic illustration of a simplified block diagram of astreaming system in accordance with an embodiment.

FIG. 3 is a schematic illustration of a simplified block diagram of adecoder in accordance with an embodiment.

FIG. 4 is a schematic illustration of a simplified block diagram of anencoder in accordance with an embodiment.

FIG. 5 is a schematic illustration of an exemplary method of encoding ordecoding in accordance with an embodiment.

FIG. 6A is graph showing the 2^(nd), 4^(th), 6^(th) and 8^(th) basis ofa 64-point DCT-2 and the 1^(st), 2^(nd), 3^(rd) and 4^(th) basis of32-point DCT-8.

FIG. 6B is a graph showing the 2^(nd), 4^(th), 6^(th) and 8^(th) basisof a 64-point DCT-2, the 1^(st), 2^(nd), 3^(rd), and 4^(th) basis ofscaled 32-point DCT-8, and the flipped 1^(st), 2^(nd), 3^(rd) and 4^(th)basis of scaled 32-point DCT-8 with inverse sign.

FIG. 7 is a schematic illustration of a computer system in accordancewith an embodiment.

DETAILED DESCRIPTION

FIG. 1 illustrates a simplified block diagram of a communication system(100) according to an embodiment of the present disclosure. The system(100) may include at least two terminals (110, 120) interconnected via anetwork (150). For unidirectional transmission of data, a first terminal(110) may code video data at a local location for transmission to theother terminal (120) via the network (150). The second terminal (120)may receive the coded video data of the other terminal from the network(150), decode the coded data and display the recovered video data.Unidirectional data transmission may be common in media servingapplications and the like.

FIG. 1 illustrates a second pair of terminals (130, 140) provided tosupport bidirectional transmission of coded video that may occur, forexample, during videoconferencing. For bidirectional transmission ofdata, each terminal (130, 140) may code video data captured at a locallocation for transmission to the other terminal via the network (150).Each terminal (130, 140) also may receive the coded video datatransmitted by the other terminal, may decode the coded data and maydisplay the recovered video data at a local display device.

In FIG. 1, the terminals (110-140) may be illustrated as a laptop 110, aserver 120, and smartphones 130 and 140, but the principles of thepresent disclosure are not so limited. Embodiments of the presentdisclosure find application with other equipment including but notlimited to laptop computers, tablet computers, media players and/ordedicated video conferencing equipment. The network (150) represents anynumber of networks that convey coded video data among the terminals(110-140), including for example wireline and/or wireless communicationnetworks. The communication network (150) may exchange data incircuit-switched and/or packet-switched channels. Representativenetworks include telecommunications networks, local area networks, widearea networks and/or the Internet. For the purposes of the presentdiscussion, the architecture and topology of the network (150) may beimmaterial to the operation of the present disclosure unless explainedherein below.

FIG. 2 illustrates, as an example for an application for the disclosedsubject matter, the placement of a video encoder and decoder in astreaming environment. The disclosed subject matter can be equallyapplicable to other video enabled applications, including, for example,video conferencing, digital TV, storing of compressed video on digitalmedia including CD, DVD, memory stick and the like, and so on.

A streaming system may include a capture subsystem (213), which caninclude a video source (201), for example a digital camera, which maycreate an uncompressed video sample stream (202). That sample stream(202), depicted as a bold line to emphasize a high data volume whencompared to encoded video bit streams, can be processed by an encoder(203) coupled to the camera (201). The encoder (203) can includehardware (e.g., a processor or circuit and memory), software, or acombination thereof to enable or implement aspects of the disclosedsubject matter as described in more detail below. The encoded video bitstream (204), depicted as a thin line to emphasize the lower data volumewhen compared to the sample stream, can be stored on a streaming server(205) for future use. One or more streaming clients (206, 208) canaccess the streaming server (205) to retrieve copies (207, 209) of theencoded video bitstream (204). A client (206) can include a videodecoder (210) which decodes the incoming copy of the encoded videobitstream (207) and creates an outgoing video sample stream (211) thatcan be rendered on a display (212) or other rendering device (notdepicted).

FIG. 3 illustrates a functional block diagram of a video decoder (210)according to an embodiment of the present disclosure. As shown in FIG.3, a receiver (310) may receive one or more codec video sequences to bedecoded by the decoder (210). In the same or another embodiment, onecoded video sequence may be received at a time, where the decoding ofeach coded video sequence is independent from other coded videosequences. The coded video sequence may be received from a channel(312), which may be a hardware/software link to a storage device whichstores the encoded video data. The receiver (310) may receive theencoded video data with other data, for example, coded audio data and/orancillary data streams, that may be forwarded to their respective usingentities (not depicted). The receiver (310) may separate the coded videosequence from the other data. To combat network jitter, a buffer memory(315) may be coupled in between receiver (310) and entropydecoder/parser (320) (“parser” henceforth). When receiver (310) isreceiving data from a store/forward device of sufficient bandwidth andcontrollability, or from an isosychronous network, the buffer (315) maynot be needed, or can be small. For use on best effort packet networkssuch as the Internet, the buffer (315) may be required, which can becomparatively large and can advantageously be of adaptive size.

The video decoder (310) may include a parser (320) to reconstructsymbols (321) from the entropy coded video sequence. Categories of thosesymbols include information used to manage operation of the decoder(210), and potentially information to control a rendering device such asa display (212) that is not an integral part of the decoder but can becoupled to it, as was shown in FIGS. 2 and 3. The control informationfor the rendering device(s) may be in the form of SupplementaryEnhancement Information (SEI messages) or Video Usability Information(VUI) parameter set fragments (not depicted). The parser (320) mayparse/entropy-decode the coded video sequence received. The coding ofthe coded video sequence can be in accordance with a video codingtechnology or standard, and can follow principles well known to a personskilled in the art, including variable length coding, Huffman coding,arithmetic coding with or without context sensitivity, and so forth. Theparser (320) may extract from the coded video sequence, a set ofsubgroup parameters for at least one of the subgroups of pixels in thevideo decoder, based upon at least one parameters corresponding to thegroup. Subgroups can include Groups of Pictures (GOPs), pictures, tiles,slices, macroblocks, Coding Units (CUs), blocks, Transform Units (TUs),Prediction Units (PUs) and so forth. The entropy decoder/parser may alsoextract from the coded video sequence information such as transformcoefficients, quantizer parameter values, motion vectors, and so forth.

The parser (320) may perform entropy decoding/parsing operation on thevideo sequence received from the buffer (315), so as to create symbols(321). Reconstruction of the symbols (321) can involve multipledifferent units depending on the type of the coded video picture orparts thereof (such as: inter and intra picture, inter and intra block),and other factors. Which units are involved, and how, can be controlledby the subgroup control information that was parsed from the coded videosequence by the parser (320). The flow of such subgroup controlinformation between the parser (320) and the multiple units below is notdepicted for clarity.

Beyond the functional blocks already mentioned, decoder 210 can beconceptually subdivided into a number of functional units as describedbelow. In a practical implementation operating under commercialconstraints, many of these units interact closely with each other andcan, at least partly, be integrated into each other. However, for thepurpose of describing the disclosed subject matter, the conceptualsubdivision into the functional units below is appropriate.

A first unit may be the scaler/inverse transform unit (351). Thescaler/inverse transform unit (351) may receive a quantized transformcoefficient as well as control information, including which transform touse, a block size, a quantization factor, quantization scaling matrices,etc. as symbol(s) (321) from the parser (320). It can output blockscomprising sample values that can be input into an aggregator (355).

In some cases, the output samples of the scaler/inverse transform (351)can pertain to an intra-coded block, which is a block that is not usingpredictive information from previously reconstructed pictures, but canuse predictive information from previously reconstructed parts of thecurrent picture. Such predictive information can be provided by an intrapicture prediction unit (352). In some cases, the intra pictureprediction unit (352) generates a block of the same size and shape ofthe block under reconstruction, using surrounding already reconstructedinformation fetched from the current (partly reconstructed) picture(358). The aggregator (355), in some cases, adds, on a per sample basis,the prediction information the intra prediction unit (352) has generatedto the output sample information as provided by the scaler/inversetransform unit (351).

In other cases, the output samples of the scaler/inverse transform unit(351) can pertain to an inter-coded block, and a potentially motioncompensated block. In such a case, a Motion Compensation Prediction unit(353) can access reference a picture memory (357) to fetch samples usedfor prediction. After motion compensating the fetched samples inaccordance with the symbols (321) pertaining to the block, these samplescan be added by the aggregator (355) to the output of the scaler/inversetransform unit (in this case called the residual samples or residualsignal) so to generate output sample information. The addresses withinthe reference picture memory form where the motion compensation unitfetches prediction samples can be controlled by motion vectors,available to the motion compensation unit in the form of symbols (321)that can have, for example X, Y, and reference picture components.Motion compensation also can include interpolation of sample values asfetched from the reference picture memory when sub-sample exact motionvectors are in use, motion vector prediction mechanisms, and so forth.

The output samples of the aggregator (355) can be subject to variousloop filtering techniques in the loop filter unit (356). Videocompression technologies can include in-loop filter technologies thatare controlled by parameters included in the coded video bitstream andmade available to the loop filter unit (356) as symbols (321) from theparser (320), but can also be responsive to meta-information obtainedduring the decoding of previous (in decoding order) parts of the codedpicture or coded video sequence, as well as responsive to previouslyreconstructed and loop-filtered sample values.

The output of the loop filter unit (356) can be a sample stream that canbe output to the render device (212) as well as stored in the referencepicture memory (357) for use in future inter-picture prediction.

Certain coded pictures, once fully reconstructed, can be used asreference pictures for future prediction. Once a coded picture is fullyreconstructed and the coded picture has been identified as a referencepicture (by, for example, parser (320)), the current reference picture(358) can become part of the reference picture buffer (357), and a freshcurrent picture memory can be reallocated before commencing thereconstruction of the following coded picture.

The video decoder 320 may perform decoding operations according to apredetermined video compression technology that may be documented in astandard, such as ITU-T Rec. H.265. The coded video sequence may conformto a syntax specified by the video compression technology or standardbeing used, in the sense that it adheres to the syntax of the videocompression technology or standard, as specified in the videocompression technology document or standard and specifically in theprofiles document therein. Also necessary for compliance can be that thecomplexity of the coded video sequence is within bounds as defined bythe level of the video compression technology or standard. In somecases, levels restrict the maximum picture size, maximum frame rate,maximum reconstruction sample rate (measured in, for example megasamplesper second), and/or maximum reference picture size. Limits set by levelscan, in some cases, be further restricted through Hypothetical ReferenceDecoder (HRD) specifications and metadata for HRD buffer managementsignaled in the coded video sequence.

In an embodiment, the receiver (310) may receive additional (redundant)data with the encoded video. The additional data may be included as partof the coded video sequence(s). The additional data may be used by thevideo decoder (320) to properly decode the data and/or to moreaccurately reconstruct the original video data. Additional data can bein the form of, for example, temporal, spatial, or SNR enhancementlayers, redundant slices, redundant pictures, forward error correctioncodes, and so on.

FIG. 4 may be a functional block diagram of a video encoder (203)according to an embodiment of the present disclosure.

The encoder (203) may receive video samples from a video source (201)(that is not part of the encoder) that may capture video image(s) to becoded by the encoder (203).

The video source (201) may provide the source video sequence to be codedby the encoder (203) in the form of a digital video sample stream thatcan be of any suitable bit depth (for example: 8 bit, 10 bit, 12 bit, .. . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ) and anysuitable sampling structure (for example Y CrCb 4:2:0, Y CrCb 4:4:4). Ina media serving system, the video source (201) may be a storage device,which may include a memory and processor, that stores previouslyprepared video. In a videoconferencing system, the video source (203)may include a camera that captures local image information as a videosequence. Video data may be provided as a plurality of individualpictures that impart motion when viewed in sequence. The picturesthemselves may be organized as a spatial array of pixels, wherein eachpixel can comprise one or more sample depending on the samplingstructure, color space, etc. in use. A person skilled in the art canreadily understand the relationship between pixels and samples. Thedescription below focuses on samples.

According to an embodiment, the encoder (203) may code and compress thepictures of the source video sequence into a coded video sequence (443)in real time or under any other time constraints as required by theapplication. Enforcing appropriate coding speed is one function ofcontroller (450). Controller controls other functional units asdescribed below and is functionally coupled to these units. The couplingis not depicted for clarity. Parameters set by the controller caninclude rate control related parameters (e.g., picture skip, quantizer,lambda value of rate-distortion optimization techniques), a picturesize, a group of pictures (GOP) layout, a maximum motion vector searchrange, and so forth. A person skilled in the art can readily identifyother functions of controller (450) as they may pertain to video encoder(203) optimized for a certain system design.

Some video encoders operate in what a person skilled in the art readilyrecognizes as a “coding loop.” As an oversimplified description, acoding loop can consist of the encoding part of an encoder (430)(“source coder” henceforth), which is responsible for creating symbolsbased on an input picture to be coded, and a reference picture(s), and a(local) decoder (433) embedded in the encoder (203) that reconstructsthe symbols to create the sample data a (remote) decoder also wouldcreate (as any compression between symbols and coded video bitstream islossless in the video compression technologies considered in thedisclosed subject matter). That reconstructed sample stream is input tothe reference picture memory (434). As the decoding of a symbol streamleads to bit-exact results independent of a decoder location (local orremote), the reference picture buffer content is also bit exact betweenlocal encoder and remote encoder. In other words, the prediction part ofan encoder “sees,” as reference picture samples, exactly the same samplevalues as a decoder would “see” when using prediction during decoding.This fundamental principle of reference picture synchronicity (andresulting drift, if synchronicity cannot be maintained, for example,because of channel errors) is well known to a person skilled in the art.

The operation of the “local” decoder (433) can be the same as of a“remote” decoder (210), which has already been described in detail abovein conjunction with FIG. 3. Briefly referring also to FIG. 3, however,as symbols are available and en/decoding of symbols to a coded videosequence by entropy coder (445) and parser (320) can be lossless, theentropy decoding parts of decoder (210) (including channel (312),receiver (310), buffer (315), and parser (320)) may not be fullyimplemented in local decoder (433).

An observation that can be made at this point is that any decodertechnology except the parsing/entropy decoding that is present in adecoder also necessarily needs to be present, in substantially identicalfunctional form, in a corresponding encoder. The description of encodertechnologies are the inverse of the described decoder technologies. Onlyin certain areas a more detail description is required and providedbelow.

As part of its operation, the source coder (430) may perform motioncompensated predictive coding, which codes an input frame predictivelywith reference to one or more previously-coded frames from the videosequence that were designated as “reference frames.” In this manner, thecoding engine (432) codes differences between pixel blocks of an inputframe and pixel blocks of reference frame(s) that may be selected asprediction reference(s) to the input frame.

The local video decoder (433) may decode coded video data of frames thatmay be designated as reference frames, based on symbols created by thesource coder (430). Operations of the coding engine (432) mayadvantageously be lossy processes. When the coded video data may bedecoded at a video decoder (not shown in FIG. 4), the reconstructedvideo sequence typically may be a replica of the source video sequencewith some errors. The local video decoder (433) replicates decodingprocesses that may be performed by the video decoder on reference framesand may cause reconstructed reference frames to be stored in thereference picture cache (434). In this manner, the encoder (203) maystore copies of reconstructed reference frames locally that have commoncontent as the reconstructed reference frames that will be obtained by afar-end video decoder (absent transmission errors).

The predictor (435) may perform prediction searches for the codingengine (432). That is, for a new frame to be coded, the predictor (435)may search the reference picture memory (434) for sample data (ascandidate reference pixel blocks) or certain metadata such as referencepicture motion vectors, block shapes, and so on, that may serve as anappropriate prediction reference for the new pictures. The predictor(435) may operate on a sample block-by-pixel block basis to findappropriate prediction references. In some cases, as determined bysearch results obtained by the predictor (435), an input picture mayhave prediction references drawn from multiple reference pictures storedin the reference picture memory (434).

The controller (450) may manage coding operations of the video coder(430), including, and for example, setting of parameters and subgroupparameters used for encoding the video data.

Output of all aforementioned functional units may be subjected toentropy coding in the entropy coder (445). The entropy coder maytranslate the symbols as generated by the various functional units intoa coded video sequence, by loss-less compressing the symbols accordingto technologies known to a person skilled in the art as, for exampleHuffman coding, variable length coding, arithmetic coding, and so forth.

The transmitter (440) may buffer the coded video sequence(s) as createdby the entropy coder (445) to prepare it for transmission via acommunication channel (460), which may be a hardware/software link to astorage device which would store the encoded video data. The transmitter(440) may merge coded video data from the video coder (430) with otherdata to be transmitted, for example, coded audio data and/or ancillarydata streams (sources not shown).

The controller (450) may manage operation of the encoder (203). Duringcoding, the controller (450) may assign to each coded picture a certaincoded picture type, which may affect the coding techniques that may beapplied to the respective picture. For example, pictures often may beassigned as one of the following frame types from among, for example, anIntra Picture (I picture), a Predictive picture (P picture) and aBi-directionally Predictive Picture (B Picture).

An Intra Picture (I picture) may be one that may be coded and decodedwithout using any other frame in the sequence as a source of prediction.Some video codecs allow for different types of Intra pictures,including, for example Independent Decoder Refresh Pictures. A personskilled in the art is aware of those variants of I pictures and theirrespective applications and features.

A Predictive Picture (P picture) may be one that may be coded anddecoded using intra prediction or inter prediction using at most onemotion vector and reference index to predict the sample values of eachblock.

A Bi-directionally Predictive Picture (B Picture) may be one that may becoded and decoded using intra prediction or inter prediction using atmost two motion vectors and reference indices to predict the samplevalues of each block. Similarly, multiple-predictive pictures can usemore than two reference pictures and associated metadata for thereconstruction of a single block.

Source pictures commonly may be subdivided spatially into a plurality ofsample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 sampleseach) and coded on a block-by-block basis. Blocks may be codedpredictively with reference to other (already coded) blocks asdetermined by the coding assignment applied to the blocks' respectivepictures. For example, blocks of I pictures may be codednon-predictively or they may be coded predictively with reference toalready coded blocks of the same picture (spatial prediction or intraprediction). Pixel blocks of P pictures may be coded non-predictively,via spatial prediction or via temporal prediction with reference to onepreviously coded reference pictures. Blocks of B pictures may be codednon-predictively, via spatial prediction or via temporal prediction withreference to one or two previously coded reference pictures.

The video coder (203) may perform coding operations according to apredetermined video coding technology or standard, such as ITU-T Rec.H.265. In its operation, the video coder (203) may perform variouscompression operations, including predictive coding operations thatexploit temporal and spatial redundancies in the input video sequence.The coded video data, therefore, may conform to a syntax specified bythe video coding technology or standard being used.

In an embodiment, the transmitter (440) may transmit additional datawith the encoded video. The video coder (430) may include such data aspart of the coded video sequence. Additional data may comprisetemporal/spatial/SNR enhancement layers, other forms of redundant datasuch as redundant pictures and slices, Supplementary EnhancementInformation (SEI) messages, Visual Usability Information (VUI) parameterset fragments, and so on.

As discussed above, the coding of the coded video sequence can be inaccordance with a video coding technology or standard, and can followprinciples well known to a person skilled in the art, including variablelength coding, Huffman coding, arithmetic coding with or without contextsensitivity, and so forth. In some streaming systems, the videobitstreams (204, 207, and 209) can be encoded according to certain videocoding/compression standards. An example of those standards includesITU-T HEVC Recommendation of H.265.

In HEVC, the primary transforms may have 4-point, 8-point, 16-point or32-point DCT-2 transforms core matrices, and the transform core matricesmay be represented using 8-bit integers, (i.e., an 8-bit transform corematrix). The HEVC transform core matrices of smaller DCT-2 are part oflarger DCT-2, as shown below:

4×4 DCT-2 Transform

{64, 64, 64, 64}

{83, 36,−36,−83}

{64,−64,−64, 64}

{36,−83, 83,−36}

8×8 DCT-2 Transform

{64, 64, 64, 64, 64, 64, 64, 64}

{89, 75, 50, 18,−18,−50,−75,−89}

{83, 36,−36,−83,−83,−36, 36, 83}

{75,−18,−89,−50, 50, 89, 18,−75}

{64,−64,−64, 64, 64,−64,−64, 64}

{50,−89, 18, 75,−75,−18, 89,−50}

{36,−83, 83,−36,−36, 83,−83, 36}

{18,−50, 75,−89, 89,−75, 50,−18}

16×16 DCT-2 Transform

{64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64}

{90 87 80 70 57 43 25 9 −9−25−43−57−70−80−87−90}

{89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89}

{87 57 9−43−80−90−70−25 25 70 90 80 43 −9−57−87}

{83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83}

{80 9−70−87−25 57 90 43−43−90−57 25 87 70 −9−80}

{75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75}

{70−43−87 9 90 25−80−57 57 80−25−90−9 87 43−70}

{64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64}

{57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57}

{50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50}

{43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43}

{36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36}

{25−70 90−80 43 9−57 87−87 57−9−43 80−90 70−25}

{18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18}

{9−25 43−57 70−80 87−90 90−87 80−70 57−43 25 −9}

32×32 DCT-2 Transform

{64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 6464 64 64 64 64 64 64 64}

{90 90 88 85 82 78 73 67 61 54 46 38 31 22 13 4−4−13−22−31−38−46−54−61−67−73−78−82−85−88−90−90}

{90 87 80 70 57 43 25 9−9−25−43−57−70−80−87−90−90−87−80−70−57−43−25−9 925 43 57 70 80 87 90}

{90 82 67 46 22−4−31−54−73−85−90−88−78−61−38−13 13 38 61 78 88 90 85 7354 31 4−22−46−67−82−90}

{89 75 50 18−18−50−75−89−89−75−50−18 18 50 75 89 89 75 5018−18−50−75−89−89−75−50−18 18 50 75 89}

{88 67 31−13−54−82−90−78−46−4 38 73 90 85 61 22−22−61−85−90−73−38 4 4678 90 82 54 13−31−67−88}

{87 57 9−43−80−90−70−25 25 70 90 80 43−9−57−87−87−57−9 43 80 90 7025−25−70−90−80−43 9 57 87}

{85 46−13−67−90−73−22 38 82 88 54−4−61−90−78−31 31 78 90 614−54−88−82−38 22 73 90 67 13−46−85}

{83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 83 83 36−36−83−83−36 36 8383 36−36−83−83−36 36 83}

{82 22−54−90−61 13 78 85 31−46−90−67 4 73 88 38−38−88−73−4 67 9046−31−85−78−13 61 90 54−22−82}

{80 9−70−87−25 57 90 43−43−90−57 25 87 70−9−80−80−9 70 87 25−57−90−43 4390 57−25−87−70 9 80}

{78 −4−82−73 13 85 67−22−88−61 31 90 54−38−90−46 46 90 38−54−90−31 61 8822−67−85−13 73 82 4−78}

{75−18−89−50 50 89 18−75−75 18 89 50−50−89−18 75 75−18−89−50 50 8918−75−75 18 89 50−50−89−18 75}

{73−31−90−22 78 67−38−90−13 82 61−46−88−4 85 54−54−85 4 88 46−61−82 1390 38−67−78 22 90 31−73}

{70−43−87 9 90 25−80−57 57 80−25−90−9 87 43−70−70 43 87−9−90−25 8057−57−80 25 90 9−87−43 70}

{67−54−78 38 85−22−90 4 90 13−88−31 82 46−73−61 61 73−46−82 3188−13−90−4 90 22−85−38 78 54−67}

{64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 64 64−64−64 6464−64−64 64 64−64−64 64}

{61−73−46 82 31−88−13 90−4−90 22 85−38−78 54 67−67−54 78 38−85−22 904−90 13 88−31−82 46 73−61}

{57−80−25 90−9−87 43 70−70−43 87 9−90 25 80−57−57 80 25−90 9 87−43−70 7043−87−9 90−25−80 57}

{54−85 −4 88−46−61 82 13−90 38 67−78−22 90−31−73 73 31−90 22 78−67−3890−13−82 61 46−88 4 85−54}

{50−89 18 75−75−18 89−50−50 89−18−75 75 18−89 50 50−89 18 75−75−1889−50−50 89−18−75 75 18−89 50}

{46−90 38 54−90 31 61−88 22 67−85 13 73−82 4 78−78−4 82−73−13 85−67−2288−61−31 90−54−38 90−46}

{43−90 57 25−87 70 9−80 80−9−70 87−25−57 90−43−43 90−57−25 87−70−9 80−809 70−87 25 57−90 43}

{38−88 73−4−67 90−46−31 85−78 13 61−90 54 22−82 82−22−54 90−61−13 78−8531 46−90 67 4−73 88−38}

{36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 36 36−83 83−36−36 83−83 3636−83 83−36−36 83−83 36}

{31−78 90−61 4 54−88 82−38−22 73−90 67−13−46 85−85 46 13−67 90−73 2238−82 88−54−4 61−90 78−31}

{25−70 90−80 43 9−57 87−87 57−9−43 80−90 70−25−25 70−90 80−43−9 57−8787−57 9 43−80 90−70 25}

{22−61 85−90 73−38−4 46−78 90−82 54−13−31 67−88 88−67 31 13−54 82−9078−46 4 38−73 90−85 61−22}

{18−50 75−89 89−75 50−18−18 50−75 89−89 75−50 18 18−50 75−89 89−7550−18−18 50−75 89−89 75−50 18}

{13−38 61−78 88−90 85−73 54−31 4 22−46 67−82 90−90 82−67 46−22−4 31−5473−85 90−88 78−61 38−13}

{9−25 43−57 70−80 87−90 90−87 80−70 57−43 25−9 −9 25−43 57−70 80−8790−90 87−80 70−57 43−25 9}

{4−13 22−31 38−46 54−61 67−73 78−82 85−88 90−90 90−90 88−85 82−78 73−6761−54 46−38 31−22 13−4}

As shown above, each of the DCT-2 cores show symmetry/anti-symmetrycharacteristics. These symmetry/anti-symmetry characteristics include atleast the following three features (Features #1-#3). Feature #1 is thateven rows with index 0, 2, 4, . . . are symmetric with a symmetry pointbefore coefficient no. N/2. Feature #2 is that odd rows with index 1, 3,5, 7, . . . are anti-symmetric with a symmetry point before coefficientno. N/2. In addition, Feature #3 is that the N-point DCT-2 core (denotedby TN) is part of 2N-point DCT-2 (denoted by T2N) and TNx,y=T2Nx,2y,where x, y=0, 1, . . . , N−1.

Based on the above symmetry/anti-symmetry characteristics (the Features#1 and #2) and the relationship between N-point DCT-2 and 2N-point DCT-2(the Feature #3), a so-called “partial butterfly” implementation may besupported to reduce the number of operation counts (multiplications,adds/subs, shifts), and identical results of matrix multiplication canbe obtained using a partial butterfly implementation. More detailsregarding HEVC transform design can be found in Non-Patent Literature 1.

In this disclosure, if each of the transform bases is either symmetricor anti-symmetric, and the N-point transform is part of the 2N-pointtransform, it is called a “recursive transform.” Examples of a recursivetransform are DCT-2, Hadamard transform, DCT-1, DST-1, and DST-2.Reordering the transform bases of one recursive transform results inanother recursive transform.

In addition to DCT-2 and 4×4 DST-7, which have been employed in HEVC, anAdaptive Multiple Transform (AMT, or also known as Enhanced MultipleTransform (EMT), or also known as Multiple Transform Selection (MTS))scheme has been proposed for residual coding for both inter and intracoded blocks. The AMT may use multiple selected transforms from theDCT/DST families other than the current transforms in HEVC. The newlyintroduced transform matrices may include DST-7, DCT-8, DST-1 and DCT-5.

Table 1 below shows exemplary basis functions of a selected DST/DCT.

TABLE 1 Transform basis functions of DCT-II/V/VIII and DST-I/VII forN-point input Transform Type Basis function T_(i)(j), i, j = 0, 1, . . ., N − 1 DCT-2${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-5${{T_{i}(j)} = {\omega_{0} \cdot \omega_{1} \cdot \sqrt{\frac{2}{{2N} - 1}} \cdot {\cos\left( \frac{2{\pi \cdot i \cdot j}}{{2N} - 1} \right)}}},$${{where}\mspace{14mu}\omega_{0}} = \left\{ {\begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix},{\omega_{1} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {j = 0} \\1 & {j \neq 0}\end{matrix} \right.}} \right.$ DCT-8${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}}$DST-1${T_{i}(j)} = {\sqrt{\frac{2}{N + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {i + 1} \right) \cdot \left( {j + 1} \right)}{N + 1} \right)}}$DST-7${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

In order to keep the orthogonality of the transform matrix, thetransform matrices may be quantized more accurately than the transformmatrices in HEVC, with 10-bit representation instead of 8-bit in HEVC.To keep the intermediate values of the transformed coefficients withinthe range of 16-bit, after horizontal and after vertical transform, allthe coefficients may be right shifted by 2 more bits compared to theright shift used in the current HEVC transforms. The AMT may apply tocoding units (CUs) with both width and height smaller than or equal to64, and whether AMT applies or not may be controlled by a CU level flag.When the CU level flag is equal to 0, DCT-2 may be applied in the CU toencode the residue. For luma coding block within an AMT-enabled CU, twoadditional flags may be signalled to identify the horizontal andvertical transform to be used.

For intra residue coding, due to the different residual statistics ofdifferent intra prediction modes, a mode-dependent transform candidateselection process may be used. Three transform subsets may be defined asshown in Table 2 below, and the transform subset may be selected basedon the intra prediction mode, as specified in Table 3 below.

TABLE 2 Three pre-defined transform candidate sets Transform SetTransform Candidates 0 DST-7 DCT-8 1 DST-7, DST-1 2 DST-7, DCT-5

With the subset concept, a transform subset may be first identifiedbased on Table 3 using the intra prediction mode of a CU with a CU-levelAMT flag that is equal to 1. After that, for each of the horizontaltransform and the vertical transform, one of the two transformcandidates in the identified transform subset, according to Table 2, maybe selected based on being explicitly signalled with flags.

TABLE 3 Selected (H)orizontal and (V)ertical transform sets for eachintra prediction mode Intra Mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1516 17 V 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 H 2 1 0 1 0 1 0 1 0 1 0 1 01 2 2 2 2 Intra Mode 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34V 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 H 2 2 2 2 2 1 0 1 0 1 0 1 0 1 0 1 0Intra Mode 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 V 1 0 10 1 0 1 0 1 0 1 2 2 2 2 2 2 2 H 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0Intra Mode 53 54 55 56 57 58 59 60 61 62 63 64 65 66 V 2 2 1 0 1 0 1 0 10 1 0 1 0 H 0 0 1 0 1 0 1 0 1 0 1 0 1 0

However, for inter prediction residual, only one transform set, whichconsists of DST-7 and DCT-8, may be used for all inter modes and forboth horizontal and vertical transforms. Among the four additionaltransform types, i.e., DST-7, DCT-8, DST-1 and DCT-5, the mostefficiently used transform types are DST-7 and DCT-8. It is noted thatDCT-8 is essentially flipping DST-7 basis left and right with signchanges, so DCT-8 and DST-7 may basically share the same transformbases.

The transform core, which is a matrix composed by the basis vectors, ofDST-7 can be also be represented as shown below:

4-Point DST-7:

a, b, c, d

c, c, 0, −c

d, −a, −c, b,

b, −d, c, −a

8-Point DST-7:

a, b, c, d, e, f, g, h

c, f, h, e, b, −a, −d, −g

e, g, b, c, −h, −d, a, f

g, c, −d, −f, a, h, b, −e

h, −a, −g, b, f, −c, e, d

d, −h, e, −a, −c, g, −f, b

b, −d, f, −h, g, −e, c, −a

16-Point DST-7:

a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p

c, f, i, l, o, o, l, i, f, c, 0, −c, −f, −i, −l, −o

e, j, o, m, h, c, −b, −g, −l, −p, −k, −f, −a, d, i, n

g, n, l, e, −b, −i, −p, −j, −c, d, k, o, h, a, −f, −m

i, o, f, −c, −l, −l, −c, f, o, i, 0, −i, −o, −f, c, l

k, k, 0, −k, −k, 0, k, k, 0, −k, −k, 0, k, k, 0, −k

m, g, −f, −n, −a, l, h, −e, −o, −b, k, i, −d, −p, −c, j

o, c, −l, −f, i, i, −f, −l, c, o, 0, −o, −c, l, f, −i

p, −a, −o, b, n, −c, −m, d, l, −e, −k, f, j, −g, −i, h

n, −e, −i, j, d, −o, a, m, −f, −h, k, c, −p, b, l, −g

l, −i, −c, o, −f, −f, o, −c, −i, l, 0, −l, i, c, −o, f

j, −m, c, g, −p, f, d, −n, i, a, −k, l, −b, −h, o, −e

h, −p, i, −a, −g, o, −j, b, f, −n, k, −c, −e, m, −l, d

f, −l, o, −i, c, c, −i, o, −l, f, 0, −f, l, −o, i, −c

d, −h, l, −p, m, −i, e, −a, −c, g, −k, o, −n, j, −f, b

b, −d, f, −h, j, −l, n, −p, o, −m, k, −i, g, −e, c, −a

32-Point DST-7:

a, b, c, d, e, f, g, h, i, j, k l, m, n, o, p, q, r, s, t, u, v, w, x,y, z, A, B, C, D, E, F

c, f, i, l, o, r, u, x, A, D, F, C, z, w, t, q, n, k, h, e, b, −a, −d,−g, −j, −m, −p, −s, −v, −y, −B, −E

e, j, o, t, y, D, D, y, t, o, j, e, 0, −e, −j, −o, −t, −y, −D, −D, −y,−t, −o, −j, −e, 0, e, j, o, t, y, D

g, n, u, B, D, w, p, i, b, −e, −l, −s, −z, −F, −y, −r, −k, −d, c, j, q,x, E, A, t, m, f, −a, −h, −c, −v, −C

i, r, A, C, t, k, b, −g, −p, −y, −E, −v, −m, −d, e, n, w, F, x, o, f,−c, −l, −u, −D, −z, −q, −h, a, j, s, B

k, v, F, u, j, −a, −l, −w, −E, −t, −i, b, m, x, D, s, h, −c, −n, −y, −C,−r, −g, d, o, z, B, q, f, −e, −p, −A

m, z, z, m, 0, −m, −z, −z, −m, 0, −m, −z, −m, 0, m, z, z, m, 0, −m, −z,−z, −m, e, m, z, z, m, 0, −m, −z, −z, −m, 0, m z

o, D, t, e, −j, −y, −y, −j, e, t, D, o, 0, −o, −D, −t, −e, j, y, y, j,−e, −t, −D, −o, 0, o, D, t, e, −j, −y,

q, E, n, −o, −t, −N, −k, f, w, y, h, −i, −z, −v, −e, l, C, s, b, −o, −F,−p, a, r, D, m, −d, −u, −A, −j, g, x

s, A, h, −k, −D, −p, o, v, x, e, −n, −F, −m, f, y, n, b, −q, −C, −j, i,B, r, −a, −t, −z, −g, l, E, o, −d, −w

u, w, b, −s, −y, −d, q, A, f, −o, −C, −h, m, E, j, −k, −F, −l, i, D, n,−g, −B, −p, s, z, r, −c, −x, −t, a, v

w, s, −d, −A, −o, h, E, k, −l, −D, −g, p, z, c, −t, −v, a, x, r, −e, −B,−n, i, F, j, −m, −C, −f, q, y, b, −u

y, o, −j, −D, −e, t, t, −e, D, −j, c, y, 0, −y, −o, j, D, e, t, −t, o,D, j, −o, −y, 0, y, o, −j, −D, −e, t

A, k, −p, −v, e, F, f, −u, −q, j, D, a, −z, −l, o, w, −d, −E, −g, t, r,−i, −C, −b, y, m, −D, −x, c, D, h, −c

C, g, −v, −n, o, u, −h, −B, e, D, f, −w, −m, p, t, −i, −A, b, E, e, −x,−l, q, s, −j, −z, c, F, d, −y, −k, r

E, o, −B, −f, y, i, −v, −l, s, o, −p, −r, m, u, −j, −x, g, A, −d, −D, e,F, b, −C, −e, z, h, −w, −k, t, n, −q

F, −a, −E, b, D, −c, −C, d, B, −e, −A, f, z, −g, −y, h, x, −i, −w, j, v,−k, −u, l, t, −m, −s, n, r, −o, −q, p

D, −e, −y, j, t, −o, −o, t, j, −y, −e, D, o, −D, e, y, −j, −y, o, o, −t,−j, y, e, −D, 0, D, −e, −y, j, t, −o

B, −i, −s, r, j, −A, −a, C, −h, −t, q, k, −z, −b, D, −g, −u, p, l, −y,−c, E, −f, −v, o, m, −x, −d, F, −e, −w, n

z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z, 0, z,−m, −m, z, 0, −z, m, m, −z, 0, z, −m

x, −q, −g, E, −j, −n, A, −c, −u, t, d, −B, m, k, −D, f, r, −w, −a, y,−p, −h, F, −i, −o, z, −b, −v, e, e, −C, l

v, −u, −e, w, −t, −b, x, −n, −c, y, −r, −d, z, −q, −e, A, −p, −f, B, −o,−g, C, −n, −h, D, −m, −i, E, −l, −j, F, −k

t, −y, e, c, −D, j, j, −D, o, e, −y, t, 0, −t, y, −e, −o, D, −j, −j, D,−o, −e, y, −t, 0, t, −y, e, c, −D, j

r, −C, k, g, −y, v, −d, −n, F, −o, −c, u, −z, h, j, −B, s, −a, −q, D,−l, −f, x, −w, e, m, −E, p, b, −t, A, −i

p, −F, q, −a, −o, E, −r, b, n, −D, s, −c, −m, C, −t, d, l, −B, u, −e,−k, A, −v, f, j, −z, w, −g, −i, y, −x, h

n, −B, w, −i, −o, s, −F, r, −d, −j, −x, −A, m, a, −o, C, −v, h, f, −t,E, −q, c, k, −y, z, −l, −b, p, −D, u, −g

l, −x, C, −q, o, g, −c, E, −v, j, b, −n, z, −A, o, −c, −i, u, −F, t, −h,−d, p, −B, y, −m, a, k, −w, D, −r, f

j, −t, D, −y, o, −e, −e, o, −y, D, −t, j, 0, −j, t, −D, y, −o, e, e, −o,y, −D, t, −j, 0, j, −t, D, −y, o, −e

h, −p, x, −F, y, −q, i, −a, −g, o, −w, E, −z, r, −j, b, f, −n, v, −D, A,−s, k, −c, −e, m, −u, C, −B, t, −l, d

f, −l, r, −x, D, −C, w, −q, k, −e, −a, g, −m, s, −y, E, −B, v, −p, j,−d, −b, h, −n, t, −x, F, −A, u, −c, i, −c

d, −h, l, −p, t, −x, B, −F, C, −y, u, −q, m, −i, e, −a, −c, q, −k, o,−o, w, −A, E, −D, z, −v, r, −n, j, −f, b

b, −d, f, −h, j, −l, n, −p, r, −t, v, −x, z, −B, D, −F, E, −C, A, −y, w,−u, s, −q, o, −m, k, −i, g, −e, c, −a

64-Point DST-7:

{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl,}

{ab, ae, ah, ak, an, aq, at, aw, az, bc, bf, bi, bl, bo, br, bu, bx, ca,cd, cg, cj, cl, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh, be, bb, ay, av,as, ap, am, aj, ag, ad, aa, −ac, −af, −ai, −al, −ao, −ar, −au, −ax, −ba,−bd, −bg, −bj, −bm, −bp, −bs, −bv, −by, −cb, −ce, −ch, −ck,}

{ac, ah, am, ar, aw, bb, bg, bl, bq, by, ca, cf, ck, ci, cd, by, bt, bo,bj, be, az, au, ap, ak, af, aa, −ae, −aj, −ao, −at, −ay, −bd, −bi, −bn,−bs, −bx, −cc, −ch, −cl, −cg, −cb, −bw, −br, −bm, −bh, −bc, −ax, −as,−an, −ai, −ad, ab, ag, al, aq, av, ba, bf, bk, bp, bu, bz, ce, cj,}

{ad, ak, ar, ay, bf, bm, bt, ca, ch, cj, cc, by, bo, bh, ba, at, am, af,−ab, −ai, −ap, −aw, −bd, −bk, −br, −by, −cf, −cl, −ce, −bx, −bq, −bj,−bc, −ay, −ao, −ah, −aa, ag, an, au, bb, bi, bp, bw, cd, ck, cg, bz, bs,bl, be, ax, aq, aj, ac, −ae, −al, −as, −az, −bg, −bn, −bu, −cb, −ci,}

{ae, an, aw, bf, bo, bx, cg, ci, bz, bq, bh, ay, ap, ag, −ac, −al, −au,−bd, −bm, −by, −ce, −ck, −cb, −bs, −bj, −ba, −ar, −ai, aa, aj, as, bb,bk, bt, cc, cl, cd, bu, bl, bc, at, ak, ab, −ah, −aq, −az, −bi, −br,−ca, −cj, −cf, −bw, −bn, −be, −ay, −am, −ad, af, ao, ax, bg, bp, by,ch,}

{af, aq, bb, bm, bx, ci, ce, bt, bi, ax, am, ab, −aj, −au, −bf, −bq,−cb, −cl, −ca, −bp, −be, −at, −ai, ac, an, ay, bj, bu, cf, ch, bw, bl,ba, ap, ae, −ag, −ar, −bc, −bn, −by, −cj, −cd, −bs, −bh, −aw, −al, −aa,ak, ay, bg, br, cc, ck, bz, bo, bd, as, ah, −ad, −ao, −az, −bk, −by,−cg,}

{ag, at, bg, bt, cg, ce, br, be, ar, ae, −ai, −ay, −bi, −by, −ci, −cc,−bp, −bc, −ap, −ac, ak, ax, bk, bx, ck, ca, bn, ba, an, aa, −am, −az,−bm, −bz, −cl, −by, −bl, −ay, −al, ab, ao, bb, bo, cb, cj, bw, bj, aw,aj, −ad, −aq, −bd, −bq, −cd, −ch, −bu, −bh, −au, −ah, af, as, bf, bs,cf,}

{ah, aw, bl, ca, ci, bt, be, ap, aa, −ao, −bd, −bs, −ch, −cb, −bm, −ax,−ai, ag, ay, bk, bz, cj, bu, bf, aq, ab, −an, −bc, −br, −cg, −cc, −bn,−ay, −aj, af, au, bj, by, ck, by, bg, ar, ac, −am, −bb, −bq, −cf, −cd,−bo, −az, −ak, ae, at, bi, bx, cl, bw, bh, as, ad, −al, −ba, −bp, −ce,}

{ai, az, bq, ch, bz, bi, ar, aa, −aq, −bh, −by, −ci, −br, −ba, −aj, ah,ay, bp, cg, ca, bj, as, ab, −ap, −bg, −bx, −cj, −bs, −bb, −ak, ag, ax,bo, cf, cb, bk, at, ac, −ao, −bf, −bw, −ck, −bt, −bc, −al, af, aw, bn,ce, cc, bl, au, ad, −an, −be, −by, −cl, −bu, −bd, −am, ae, av, bm, cd,}

{aj, bc, by, cj, bq, ax, ae, −ao, −bh, −ca, −ce, −bl, −as, aa, at, bm,cf, bz, bg, an, −af, −ay, −br, −ck, −bu, −bb, −ai, ak, bd, bw, ci, bp,aw, ad, −ap, −bi, −cb, −cd, −bk, −ar, ab, au, bn, cg, by, bf, am, −ag,−az, −bs, −cl, −bt, −ba, −ah, al, be, bx, ch, bo, av, ac, −aq, −bj,−cc,}

{ak, bf, ca, cc, bh, am, −ai, −bd, −by, −ce, −bj, −ao, ag, bb, bw, cg,bl, aq, −ae, −az, −bu, −ci, −bn, −as, ac, ax, bs, ck, bp, au, −aa, −av,−bq, −cl, −br, −aw, −ab, at, bo, cj, bt, ay, ad, −ar, −bm, −ch, −by,−ba, −af, ap, bk, cf, bx, bc, ah, −an, −bi, −cd, −bz, −be, −aj, al, bg,cb,}

{al, bi, cf, by, ay, ab, −ay, −bs, −ci, −bl, −ao, ai, bf, cc, by, bb,ae, −as, −bp, −cl, −bo, −ar, af, bc, bz, cb, be, ah, −ap, −bm, −cj, −br,−au, ac, az, bw, ce, bh, ak, −am, −bj, −cg, −bu, −ax, −aa, aw, bt, ch,bk, an, −aj, −bg, −cd, −bx, −ba, −ad, at, bq, ck, bn, aq, −ag, −bd,−ca,}

{am, bl, ck, bo, ap, −aj, −bi, −ch, −br, −as, ag, bf, ce, bu, ay, −ad,−bc, −cb, −bx, −ay, aa, az, by, ca, bb, ac, −aw, −by, −cd, −be, −af, at,bs, cg, bh, ai, −aq, −bp, −cj, −bk, −al, an, bm, cl, bn, ao, −ak, −bj,−ci, −bq, −ar, ah, bg, cf, bt, au, −ae, −bd, −cc, −bw, −ax, ab, ba, bz,}

{an, bo, ci, bh, ag, −au, −by, −cb, −ba, aa, bb, cc, bu, at, −ah, −bi,−cj, −bn, −am, ao, bp, ch, bg, af, −ay, −bw, −ca, −az, ab, bc, cd, bt,as, −ai, −bj, −ck, −bm, −al, ap, bq, cg, bf, ae, −aw, −bx, −bz, −ay, ac,bd, ce, bs, ar, −aj, −bk, −cl, −bl, −ak, aq, br, cf, be, ad, −ax, −by,}

{ao, br, cd, ba, −ac, −bf, −ci, −bm, −aj, at, bw, by, ay, −ah, −bk, −ck,−bh, −ae, ay, cb, bt, aq, −am, −bp, −cf, −bc, aa, bd, cg, bo, al, −ar,−bu, −ca, −ax, af, bi, cl, bj, ag, −aw, −bz, −by, −as, ak, bn, ch, be,ab, −bb, −ce, −bq, −an, ap, bs, cc, az, −ad, −bg, −cj, −bl, −ai, au,bx,}

{ap, bu, by, at, −al, −bq, −cc, −ax, ah, bm, cg, bb, −ad, −bi, −ck, −bf,−aa, be, cj, bj, ae, −ba, −cf, −bn, −ai, aw, cb, br, am, −as, −bx, −by,−aq, ao, bt, bz, au, −ak, −bp, −cd, −ay, ag, bl, ch, bc, −ac, −bh, −cl,−bg, −ab, bd, ci, bk, af, −az, −ce, −bo, −aj, av, ca, bs, an, −ar, −bw,}

{aq, bx, bt, am, −au, −cb, −bp, −ai, ay, cf, bl, ae, −bc, −cj, −bh, −aa,bg, ck, bd, −ad, −bk, −cg, −az, ah, bo, cc, av, −al, −bs, −by, −ar, ap,bw, bu, an, −at, −ca, −bq, −aj, ax, ce, bm, af, −bb, −ci, −bi, −ab, bf,cl, be, −ac, −bj, −ch, −ba, ag, bn, cd, aw, −ak, −br, −bz, −as, ao, bv,}

{ar, ca, bo, af, −bd, −cl, −bc, ag, bp, bz, aq, −as, −cb, −bn, −ae, be,ck, bb, −ah, −bq, −by, −ap, at, cc, bm, ad, −bf, −cj, −ba, ai, br, bx,ao, −au, −cd, −bl, −ac, bg, ci, az, −aj, −bs, −bw, −an, av, ce, bk, ab,−bh, −ch, −ay, ak, bt, by, am, −aw, −cf, −bj, −aa, bi, cg, ax, −al,−bu,}

{as, cd, bj, −ab, −bm, −ca, −ap, av, cg, bg, −ae, −bp, −bx, −am, ay, cj,bd, −ah, −bs, −bu, −aj, bb, cl, ba, −ak, −by, −br, −ag, be, ci, ax, −an,−by, −bo, −ad, bh, cf, au, −aq, −cb, −bl, −aa, bk, cc, ar, −at, −ce,−bi, ac, bn, bz, ao, −aw, −ch, −bf, af, bq, bw, al, −az, −ck, −bc, ai,bt,}

{at, cg, be, −ai, −by, −bp, −ac, bk, ca, an, −az, −cl, −ay, ao, cb, bj,−ad, −bq, −bu, −ah, bf, cf, as, −au, −ch, −bd, aj, bw, bo, ab, −bl, −bz,−am, ba, ck, ax, −ap, −cc, −bi, ae, br, bt, ag, −bg, −ce, −ar, ay, ci,bc, −ak, −bx, −bn, −aa, bm, by, al, −bb, −cj, −aw, aq, cd, bh, −af,−bs,}

{au, cj, az, −ap, −ce, −be, ak, bz, bj, −af, −bu, −bo, aa, bp, bt, ae,−bk, −by, −aj, bf, cd, ao, −ba, −ci, −at, ay, ck, ay, −aq, −cf, −bd, al,ca, bi, −ag, −by, −bn, ab, bq, bs, ad, −bl, −bx, −ai, bg, cc, an, −bb,−ch, −as, aw, cl, ax, −ar, −cg, −bc, am, cb, bh, −ah, −bw, −bm, ac, br,}

{av, cl, au, −aw, −ck, −at, ax, cj, as, −ay, −ci, −ar, az, ch, aq, −ba,−cg, −ap, bb, cf, ao, −bc, −ce, −an, bd, cd, am, −be, −cc, −al, bf, cb,ak, −bg, −ca, −aj, bh, bz, ai, −bi, −by, −ah, bj, bx, ag, −bk, −bw, −af,bl, by, ae, −bm, −bu, −ad, bn, bt, ac, −bo, −bs, −ab, bp, br, aa, −bq,}

{aw, ci, ap, −bd, −cb, −ai, bk, bu, ab, −br, −bn, af, by, bg, −am, −cf,−az, at, cl, as, −ba, −ce, −al, bh, bx, ae, −bo, −bq, ac, by, bj, −aj,−cc, −bc, aq, cj, av, −ax, −ch, −ao, be, ca, ah, −bl, −bt, −aa, bs, bm,−ag, −bz, −bf, an, cg, ay, −au, −ck, −ar, bb, cd, ak, −bi, −bw, −ad,bp,}

{ax, cf, ak, −bk, −bs, ac, bx, bf, −ap, −ck, −as, bc, ca, af, −bp, −bn,ah, cc, ba, −au, −ci, −an, bh, by, aa, −bu, −bi, am, ch, av, −az, −cd,−ai, bm, bq, −ae, −bz, −bd, ar, cl, aq, −be, −by, −ad, br, bl, −aj, −ce,−ay, aw, cg, al, −bj, −bt, ab, bw, bg, −ao, −cj, −at, bb, cb, ag, −bo,}

{ay, cc, af, −br, −bj, an, ck, aq, −bg, −bu, ac, bz, bb, −av, −cf, −ai,bo, bm, −ak, −ch, −at, bd, bx, aa, −bw, −be, as, ci, al, −bl, −bp, ah,ce, aw, −ba, −ca, −ad, bt, bh, −ap, −cl, −ao, bi, bs, −ae, −cb, −az, ax,cd, ag, −bq, −bk, am, cj, ar, −bf, −by, ab, by, bc, −au, −cg, −aj, bn,}

{az, bz, aa, −by, −ba, ay, ca, ab, −bx, −bb, ax, cb, ac, −bw, −bc, aw,cc, ad, −by, −bd, ay, cd, ae, −bu, −be, au, ce, af, −bt, −bf, at, cf,ag, −bs, −bg, as, cg, ah, −br, −bh, ar, ch, ai, −bq, −bi, aq, ci, aj,−bp, −bj, ap, cj, ak, −bo, −bk, ao, ck, al, −bn, −bl, an, cl, am, −bm,}

{ba, bw, −ae, −cf, −ar, bj, bn, −an, −cj, −ai, bs, be, −aw, −ca, aa, cb,ay, −bf, −br, aj, ck, am, −bo, −bi, as, ce, ad, −bx, −az, bb, by, −af,−cg, −aq, bk, bm, −ao, −ci, −ah, bt, bd, −ax, −bz, ab, cc, au, −bg, −bq,ak, cl, al, −bp, −bh, at, cd, ac, −by, −ay, bc, bu, −ag, −ch, −ap, bl,}

{bb, bt, −aj, −cl, −ai, bu, ba, −bc, −bs, ak, ck, ah, −by, −az, bd, br,−al, −cj, −ag, bw, ay, −be, −bq, am, ci, af, −bx, −ax, bf, bp, −an, −ch,−ae, by, aw, −bg, −bo, ao, cg, ad, −bz, −ay, bh, bn, −ap, −cf, −ac, ca,au, −bi, −bm, aq, ce, ab, −cb, −at, bj, bl, −ar, −cd, −aa, cc, as, −bk,}

{bc, bq, −ao, −ce, aa, cf, an, −br, −bb, bd, bp, −ap, −cd, ab, cg, am,−bs, −ba, be, bo, −aq, −cc, ac, ch, al, −bt, −az, bf, bn, −ar, −cb, ad,ci, ak, −bu, −ay, bg, bm, −as, −ca, ae, cj, aj, −by, −ax, bh, bl, −at,−bz, af, ck, ai, −bw, −aw, bi, bk, −au, −by, ag, cl, ah, −bx, −ay, bj,}

{bd, bn, −at, −bx, aj, ch, aa, −cg, −ak, bw, au, −bm, −be, bc, bo, −as,−by, ai, ci, ab, −cf, −al, by, av, −bl, −bf, bb, bp, −ar, −bz, ah, cj,ac, −ce, −am, bu, aw, −bk, −bg, ba, bq, −aq, −ca, ag, ck, ad, −cd, −an,bt, ax, −bj, −bh, az, br, −ap, −cb, af, cl, ae, −cc, −ao, bs, ay, −bi,}

{be, bk, −ay, −bq, as, bw, −am, −cc, ag, ci, −aa, −cj, −af, cd, al, −bx,−ar, br, ax, −bl, −bd, bf, bj, −az, −bp, at, by, −an, −cb, ah, ch, −ab,−ck, −ae, ce, ak, −by, −aq, bs, aw, −bm, −bc, bg, bi, −ba, −bo, au, bu,−ao, −ca, ai, cg, −ac, −cl, −ad, cf, aj, −bz, −ap, bt, av, −bn, −bb,bh,}

{bf, bh, −bd, −bj, bb, bl, −az, −bn, ax, bp, −av, −br, at, bt, −ar, −by,ap, bx, −an, −bz, al, cb, −aj, −cd, ah, cf, −af, −ch, ad, cj, −ab, −cl,−aa, ck, ac, −ci, −ae, cg, ag, −ce, −ai, cc, ak, −ca, −am, by, ao, −bw,−aq, bu, as, −bs, −au, bq, aw, −bo, −ay, bm, ba, −bk, −bc, bi, be, −bg,}

{bg, be, −bi, −bc, bk, ba, −bm, −ay, bo, aw, −bq, −au, bs, as, −bu, −aq,bw, ao, −by, −am, ca, ak, −cc, −ai, ce, ag, −cg, −ae, ci, ac, −ck, −aa,cl, −ab, −cj, ad, ch, −af, −cf, ah, cd, −aj, −cb, al, bz, −an, −bx, ap,by, −ar, −bt, at, br, −ay, −bp, ax, bn, −az, −bl, bb, bj, −bd, −bh, bf,}

{bh, bb, −bn, −ay, bt, ap, −bz, −aj, cf, ad, −cl, ac, cg, −ai, −ca, ao,bu, −au, −bo, ba, bi, −bg, −bc, bm, aw, −bs, −aq, by, ak, −ce, −ae, ck,−ab, −ch, ah, cb, −an, −by, at, bp, −az, −bj, bf, bd, −bl, −ax, br, ar,−bx, −al, cd, af, −cj, aa, ci, −ag, −cc, am, bw, −as, −bq, ay, bk, −be,}

{bi, ay, −bs, −ao, cc, ae, −cl, af, cb, −ap, −br, az, bh, −bj, −ax, bt,an, −cd, −ad, ck, −ag, −ca, aq, bq, −ba, −bg, bk, aw, −bu, −am, ce, ac,−cj, ah, bz, −ar, −bp, bb, bf, −bl, −ay, by, al, −cf, −ab, ci, −ai, −by,as, bo, −bc, −be, bm, au, −bw, −ak, cg, aa, −ch, aj, bx, −at, −bn, bd,}

{bj, ay, −bx, −ah, cl, −ag, −by, au, bk, −bi, −aw, bw, ai, −ck, af, bz,−at, −bl, bh, ax, −by, −aj, cj, −ae, −ca, as, bm, −bg, −ay, bu, ak, −ci,ad, cb, −ar, −bn, bf, az, −bt, −al, ch, −ac, −cc, aq, bo, −be, −ba, bs,am, −cg, ab, cd, −ap, −bp, bd, bb, −br, −an, cf, −aa, −ce, ao, bq, −bc,}

{bk, as, −cc, −aa, cd, −ar, −bl, bj, at, −cb, −ab, ce, −aq, −bm, bi, au,−ca, −ac, cf, −ap, −bn, bh, av, −bz, −ad, cg, −ao, −bo, bg, aw, −by,−ae, ch, −an, −bp, bf, ax, −bx, −af, ci, −am, −bq, be, ay, −bw, −ag, cj,−al, −br, bd, az, −by, −ah, ck, −ak, −bs, bc, ba, −bu, −ai, cl, −aj,−bt, bb,}

{bl, ap, −ch, ag, bu, −bc, −ay, by, ac, −cd, at, bh, −bp, −al, cl, −ak,−bq, bg, au, −cc, ab, bz, −ax, −bd, bt, ah, −ci, ao, bm, −bk, −aq, cg,−af, −by, bb, az, −bx, −ad, ce, −as, −bi, bo, am, −ck, aj, br, −bf, −av,cb, −aa, −ca, aw, be, −bs, −ai, cj, −an, −bn, bj, ar, −cf, ae, bw, −ba,}

{bm, am, −cl, an, bl, −bn, −al, ck, −ao, −bk, bo, ak, −cj, ap, bj, −bp,−aj, ci, −aq, −bi, bq, ai, −ch, ar, bh, −br, −ah, cg, −as, −bg, bs, ag,−cf, at, bf, −bt, −af, ce, −au, −be, bu, ae, −cd, av, bd, −by, −ad, cc,−aw, −bc, bw, ac, −cb, ax, bb, −bx, −ab, ca, −ay, −ba, by, aa, −bz, az,}

{bn, aj, −cg, au, bc, −by, ab, by, −bf, −ar, cj, −am, −bk, bq, ag, −cd,ax, az, −cb, ae, bs, −bi, −ao, cl, −ap, −bh, bt, ad, −ca, ba, aw, −ce,ah, bp, −bl, −al, ci, −as, −be, bw, aa, −bx, bd, at, −ch, ak, bm, −bo,−ai, cf, −ay, −bb, bz, −ac, −bu, bg, aq, −ck, an, bj, −br, −af, cc,−ay,}

{bo, ag, −cb, bb, at, −cj, ao, bg, −bw, ab, bt, −bj, −al, cg, −aw, −ay,ce, −aj, −bl, br, ad, −by, be, aq, −cl, ar, bd, −bz, ae, bq, −bm, −ai,cd, −az, −ay, ch, −am, −bi, bu, aa, −by, bh, an, −ci, au, ba, −cc, ah,bn, −bp, −af, ca, −bc, −as, ck, −ap, −bf, bx, −ac, −bs, bk, ak, −cf,ax,}

{bp, ad, −bw, bi, ak, −cd, bb, ar, −ck, au, ay, −cg, an, bf, −bz, ag,bm, −bs, −aa, bt, −bl, −ah, ca, −be, −ao, ch, −ax, −ay, cj, −aq, −bc,cc, −aj, −bj, by, −ac, −bq, bo, ae, −bx, bh, al, −ce, ba, as, −cl, at,az, −cf, am, bg, −by, af, bn, −br, −ab, bu, −bk, −ai, cb, −bd, −ap, ci,−aw,}

{bq, aa, −br, bp, ab, −bs, bo, ac, −bt, bn, ad, −bu, bm, ae, −by, bl,af, −bw, bk, ag, −bx, bj, ah, −by, bi, ai, −bz, bh, aj, −ca, bg, ak,−cb, bf, al, −cc, be, am, −cd, bd, an, −ce, bc, ao, −cf, bb, ap, −cg,ba, aq, −ch, az, ar, −ci, ay, as, −cj, ax, at, −ck, aw, au, −cl, av,}

{br, −ac, −bm, bw, −ah, −bh, cb, −am, −bc, cg, −ar, −ax, cl, −aw, −as,ch, −bb, −an, cc, −bg, −ai, bx, −bl, −ad, bs, −bq, ab, bn, −by, ag, bi,−ca, al, bd, −cf, aq, ay, −ck, av, at, −ci, ba, ao, −cd, bf, aj, −by,bk, ae, −bt, bp, −aa, −bo, bu, −af, −bj, bz, −ak, −be, ce, −ap, −az, cj,−au,}

{bs, −af, −bh, cd, −aq, −aw, cj, −bb, −al, by, −bm, −aa, bn, −bx, ak,bc, −ci, av, ar, −ce, bg, ag, −bt, br, −ae, −bi, cc, −ap, −ax, ck, −ba,−am, bz, −bl, −ab, bo, −bw, aj, bd, −ch, au, as, −cf, bf, ah, −bu, bq,−ad, −bj, cb, −ao, −ay, cl, −az, −an, ca, −bk, −ac, bp, −by, ai, be,−cg, at,}

{bt, −ai, −bc, ck, −az, −al, bw, −bq, af, bf, −ch, aw, ao, −bz, bn, −ac,−bi, ce, −at, −ar, cc, −bk, −aa, bl, −cb, aq, au, −cf, bh, ad, −bo, by,−an, −ax, ci, −be, −ag, br, −by, ak, ba, −cl, bb, aj, −bu, bs, −ah, −bd,cj, −ay, −am, bx, −bp, ae, bg, −cg, ay, ap, −ca, bm, −ab, −bj, cd, −as,}

{bu, −al, −ax, cg, −bi, −aa, bj, −cf, aw, am, −by, bt, −ak, −ay, ch,−bh, −ab, bk, −ce, ay, an, −bw, bs, −aj, −az, ci, −bg, −ac, bl, −cd, au,ao, −bx, br, −ai, −ba, cj, −bf, −ad, bm, −cc, at, ap, −by, bq, −ah, −bb,ck, −be, −ae, bn, −cb, as, aq, −bz, bp, −ag, −bc, cl, −bd, −af, bo, −ca,ar,}

{by, −ao, −as, bz, −br, ak, aw, −cd, bn, −ag, −ba, ch, −bj, ac, be, −cl,bf, ab, −bi, ci, −bb, −af, bm, −ce, ax, aj, −bq, ca, −at, −an, bu, −bw,ap, ar, −by, bs, −al, −ay, cc, −bo, ah, az, −cg, bk, −ad, −bd, ck, −bg,−aa, bh, −cj, bc, ae, −bl, cf, −ay, −ai, bp, −cb, au, am, −bt, bx, −aq,}

{bw, −ar, −an, bs, −ca, ay, aj, −bo, ce, −az, −af, bk, −ci, bd, ab, −bg,cl, −bh, ac, bc, −ch, bl, −ag, −ay, cd, −bp, ak, au, −bz, bt, −ao, −aq,by, −bx, as, am, −br, cb, −aw, −ai, bn, −cf, ba, ae, −bj, cj, −be, −aa,bf, −ck, bi, −ad, −bb, cg, −bm, ah, ax, −cc, bq, −al, −at, by, −bu, ap,}

{bx, −au, −ai, bl, −cj, bg, −ad, −az, cc, −bs, ap, an, −bq, ce, −bb,−ab, be, −ch, bn, −ak, −as, by, −bz, aw, ag, −bj, cl, −bi, af, ax, −ca,bu, −ar, −al, bo, −cg, bd, −aa, −bc, cf, −bp, am, aq, −bt, cb, −ay, −ae,bh, −ck, bk, −ah, −ay, by, −bw, at, aj, −bm, ci, −bf, ac, ba, −cd, br,−ao,}

{by, −ax, −ad, be, −cf, br, −aq, −ak, bl, −cl, bk, −aj, −ar, bs, −ce,bd, −ac, −ay, bz, −bx, aw, ae, −bf, cg, −bq, ap, al, −bm, ck, −bj, ai,as, −bt, cd, −bc, ab, az, −ca, bw, −av, −af, bg, −ch, bp, −ao, −am, bn,−cj, bi, −ah, −at, bu, −cc, bb, −aa, −ba, cb, −by, au, ag, −bh, ci, −bo,an,}

{bz, −ba, ab, ax, −bw, cc, −bd, ae, au, −bt, cf, −bg, ah, ar, −bq, ci,−bj, ak, ao, −bn, cl, −bm, an, al, −bk, cj, −bp, aq, ai, −bh, cg, −bs,at, af, −be, cd, −by, aw, ac, −bb, ca, −by, az, −aa, −ay, bx, −cb, bc,−ad, −av, bu, −ce, bf, −ag, −as, br, −ch, bi, −aj, −ap, bo, −ck, bl,−am,}

{ca, −bd, ag, aq, −bn, ck, −bq, at, ad, −ba, bx, −cd, bg, −aj, −an, bk,−ch, bt, −aw, −aa, ax, −bu, cg, −bj, am, ak, −bh, ce, −bw, az, −ac, −au,br, −cj, bm, −ap, −ah, be, −cb, bz, −bc, af, ar, −bo, cl, −bp, as, ae,−bb, by, −cc, bf, −ai, −ao, bl, −ci, bs, −av, −ab, ay, −by, cf, −bi,al,}

{cb, −bg, al, aj, −be, bz, −cd, bi, −an, −ah, bc, −bx, cf, −bk, ap, af,−ba, by, −ch, bm, −ar, −ad, ay, −bt, cj, −bo, at, ab, −aw, br, −cl, bq,−ay, aa, au, −bp, ck, −bs, ax, −ac, −as, bn, −ci, bu, −az, ae, aq, −bl,cg, −bw, bb, −ag, −ao, bj, −ce, by, −bd, ai, am, −bh, cc, −ca, bf, −ak,}

{cc, −bj, aq, ac, −ay, bo, −ch, bx, −be, al, ah, −ba, bt, −cl, bs, −az,ag, am, −bf, by, −cg, bn, −au, ab, ar, −bk, cd, −cb, bi, −ap, −ad, aw,−bp, ci, −bw, bd, −ak, −ai, bb, −bu, ck, −br, ay, −af, −an, bg, −bz, cf,−bm, at, −aa, −as, bl, −ce, ca, −bh, ao, ae, −ax, bq, −cj, by, −bc, aj,}

{cd, −bm, ay, −ae, −am, bd, −bu, cl, −by, be, −an, −ad, au, −bl, cc,−ce, bn, −aw, af, al, −bc, bt, −ck, bw, −bf, ao, ac, −at, bk, −cb, cf,−bo, ax, −ag, −ak, bb, −bs, cj, −bx, bg, −ap, −ab, as, −bj, ca, −cg, bp,−ay, ah, aj, −ba, br, −ci, by, −bh, aq, aa, −ar, bi, −bz, ch, −bq, az,−ai,}

{ce, −bp, ba, −al, −ad, as, −bh, bw, −cl, bx, −bi, at, −ae, −ak, az,−bo, cd, −cf, bq, −bb, am, ac, −ar, bg, −by, ck, −by, bj, −au, af, aj,−ay, bn, −cc, cg, −br, bc, −an, −ab, aq, −bf, bu, −cj, bz, −bk, ay, −ag,−ai, ax, −bm, cb, −ch, bs, −bd, ao, aa, −ap, be, −bt, ci, −ca, bl, −aw,ah,}

{cf, −bs, bf, −as, af, ah, −au, bh, −bu, ch, −cd, bq, −bd, aq, −ad, −aj,aw, −bj, bw, −cj, cb, −bo, bb, −ao, ab, al, −ay, bl, −by, cl, −bz, bm,−az, am, aa, −an, ba, −bn, ca, −ck, bx, −bk, ax, −ak, −ac, ap, −bc, bp,−cc, ci, −by, bi, −av, ai, ae, −ar, be, −br, ce, −cg, bt, −bg, at, −ag,}

{cg, −by, bk, −az, ao, −ad, −ah, as, −bd, bo, −bz, ck, −cc, br, −bg, av,−ak, −aa, al, −aw, bh, −bs, cd, −cj, by, −bn, bc, −ar, ag, ae, −ap, ba,−bl, bw, −ch, cf, −bu, bj, −ay, an, −ac, −ai, at, −be, bp, −ca, cl, −cb,bq, −bf, au, −aj, −ab, am, −ax, bi, −bt, ce, −ci, bx, −bm, bb, −aq, af,}

{ch, −by, bp, −bg, ax, −ao, af, ad, −am, av, −be, bn, −bw, cf, −cj, ca,−br, bi, −az, aq, −ah, −ab, ak, −at, bc, −bl, bu, −cd, cl, −cc, bt, −bk,bb, −as, aj, −aa, −ai, ar, −ba, bj, −bs, cb, −ck, ce, −by, bm, −bd, au,−al, ac, ag, −ap, ay, −bh, bq, −bz, ci, −cg, bx, −bo, bf, −aw, an, −ae,}

{ci, −cb, bu, −bn, bg, −az, as, −al, ae, ac, −aj, aq, −ax, be, −bl, bs,−bz, cg, −ck, cd, −bw, bp, −bi, bb, −au, an, −ag, −aa, ah, −ao, ay, −bc,bj, −bq, bx, −ce, cl, −cf, by, −br, bk, −bd, aw, −ap, ai, −ab, −af, am,−at, ba, −bh, bo, −by, cc, −cj, ch, −ca, bt, −bm, bf, −ay, ar, −ak, ad,}

{cj, −ce, bz, −bu, bp, −bk, bf, −ba, ay, −aq, al, −ag, ab, ad, −ai, an,−as, ax, −bc, bh, −bm, br, −bw, cb, −cg, cl, −ch, cc, −bx, bs, −bn, bi,−bd, ay, −at, ao, −aj, ae, aa, −af, ak, −ap, au, −az, be, −bj, bo, −bt,by, −cd, ci, −ck, cf, −ca, by, −bq, bl, −bg, bb, −aw, ar, −am, ah, −ac,}

{ck, −ch, ce, −cb, by, −by, bs, −bp, bm, −bj, bg, −bd, ba, −ax, au, −ar,ao, −al, ai, −af, ac, aa, −ad, ag, −aj, am, −ap, as, −ay, ay, −bb, be,−bh, bk, −bn, bq, −bt, bw, −bz, cc, −cf, ci, −cl, cj, −cg, cd, −ca, bx,−bu, br, −bo, bl, −bi, bf, −bc, az, −aw, at, −aq, an, −ak, ah, −ae, ab,}

{cl, −ck, cj, −ci, ch, −cg, cf, −ce, cd, −cc, cb, −ca, bz, −by, bx, −bw,by, −bu, bt, −bs, br, −bq, bp, −bo, bn, −bm, bl, −bk, bj, −bi, bh, −bg,bf, −be, bd, −bc, bb, −ba, az, −ay, ax, −aw, ay, −au, at, −as, ar, −aq,ap, −ao, an, −am, al, −ak, aj, −ai, ah, −ag, af, −ae, ad, −ac, ab, −aa,}

The variables a, b c, . . . aa, ab, ac, . . . , cl can be derived basedon the formulations of DST-7 shown in

Table, and their values can be different for different sizes of DST-7.For example, the value of “a” can be different for 4-point DST-7 and8-point DST-7.

To avoid floating point operations, similar to the DCT-2 cores used inHEVC, the transform core of DST-7 may be scaled by a pre-defined factor,e.g., 64·√{square root over (N)} (corresponds to 8-bit transform core)or 256·√{square root over (N)} (corresponds to 10-bit transform core),and rounded to the nearest integer, or further tuned by an offset, e.g.,+1/−1.

Different from DCT-2, of which fast methods have been extensivelystudied, the implementation of DST-7 is still much less efficient thanDCT-2, e.g., matrix multiplication based in VTM 1 of Non-PatentLiterature 2.

In Non-Patent Literature 3, a method is proposed to approximatedifferent types of DCTs and DSTs in JEM7 by applying adjustment stagesto a transform in the DCT-2 family, which includes DCT-2, DCT-3, DST-2and DST-3, and the adjustment stage refers to a matrix multiplicationusing a sparse matrix which requires relatively less operation counts.In Non-Patent Literature 4, a method for implementing N-point DST-7using 2N+1 point Discrete Fourier Transform (DFT) is proposed.

However, additional transform types in AMT increase the complexity andimplementation cost, since additional transform cores need to be storedin on-chip memory for practical implementation, and transform cores needto be loaded when switching the transform type from one block toanother. In HEVC, a smaller DCT-2 transform is a part of the largerDCT-2 transform, which supports reusing the same logic for implementingdifferent sizes of transforms. However, with AMT, such a feature ismissing, other than DCT-2, smaller transform cannot be re-used forlarger transform, which means different logic may need to be designedfor implementing different sizes of transform.

The DCT-2 in HEVC is implemented using partial butterfly, which supportsidentical outputs between matrix multiplication and partial butterfly,i.e., dual implementation. However, half-size matrix multiplication isstill needed which is still a computation burden, for example, partialbutterfly of 32-point HEVC DCT-2 requires 2×2, 4×4, 8×8 and 16×16 matrixmultiplications.

It is observed that partial of the odd bases (odd basis refers to basisvector associated with an odd index, the first basis being associatedwith index 0) in DCT-2 is very similar to the scaled DCT-8 bases. In thefollowing FIG. 6A, the second, fourth, sixth, and eighth basis vectorsof 64-point DCT-2 are shown, as indicated by solid curves. In addition,the first, second, third and fourth basis vector of 32-point DCT-8,which is scaled by a constant

$\frac{1}{\sqrt{2}},$are also shown in FIG. 6A, as indicated by dotted curves. From FIG. 6Ait can be seen that, the first half of the second, fourth, sixth, andeighth basis vectors of 64-point DCT-2 are very close to the first,second, third and fourth basis vectors of 32-point DCT-8 scaled by

$\frac{1}{\sqrt{2}}.$

Based on this observation, if we replace the first half of odd bases of2N-point DCT-2 with N-point scaled DCT-8 bases, and fill the second halfwith the flipped DCT-8 bases plus inverse signs, then the resultingbasis is very close to the original odd bases of 2N-point DCT-2, asshown in FIG. 6B.

Therefore, we can derive a new orthonormal transform, namely a CompoundOrthonormal Transform (COT), by embedding the N-point DCT-8 (or DST-7,since DST-7 and DCT-8 bases are symmetric to each other) into 2N-pointDCT-2, and the resulting transform is still an orthonormal transform dueto the symmetric/anti-symmetric characteristics of DCT-2, i.e., Feature#1 and Feature #2.

In this way, by embedding the N-point DCT-8/DST-7 into 2N-point DCT-2,the complexity of performing AMT is reduced in 3 aspects: (i) the logicof performing transform for DCT-2 and DCT-8/DST-7 can be shared, (ii)the on-chip memory cost for storing DCT-8 and DST-7 bases can be reducedsince they are part of the DCT-2 bases and (iii) with the proposed COT,lossless fast DST-7 methods from the incorporated-by-reference '076application can be applied to further reduce the computation cost ofDCT-2 since DST-7 is part of DCT-2.

The proposed methods may be used separately or combined in any order.Due to the aforementioned advantages of COT cores, it is proposed to useCOT cores instead of traditional transform cores (such as DCT-2 andDST-7) in video and/or image compression, such as transform image orresidual blocks. A 2N-point COT core may be generated by embeddingN-point orthonormal transform (with transform core being denoted by T₀,T₁, T₂ . . . ) into another 2N-point recursive transform (with transformcore being denoted by R). The resulting transform core (denoted by RT)is also an orthonormal matrix. This new resulting transform core RT (aCOT core) can be used for encoding or decoding an image block or aresidual block. A COT core has the following feature(s):

Feature #A, in one embodiment, the N-point orthonormal transform core,i.e., T₀, T₁, T₂ . . . , is replacing one half of the RT basis rowswhich are anti-symmetry, and the remaining half is derived to keep theanti-symmetry characteristics. Feature #B, in one embodiment, theN-point orthonormal transform core, i.e., T₀, T₁, T₂ . . . , isreplacing one half of the RT basis rows which are symmetry, and theremaining half is derived to keep the symmetry characteristics.

Feature #C, in one embodiment, one N-point orthonormal transform isreplacing one half of a selected set (denoted by set0) of RT basis rowswhich are symmetry, and another N-point orthonormal transform isreplacing one half of a selected set (denoted by set1) of RT basis rowswhich are anti-symmetry, and the remaining half is derived to keep theassociated symmetry and anti-symmetry characteristics. In this Feature#C example, as an example, set0 refers to the RT basis rows locatedwithin the last N rows of RT and set1 refers to the RT basis rowslocated within the first N rows of RT.

Feature #D, in one embodiment, RT is a transform from the DCT/DSTfamilies, examples are: DCT-2, DCT-3, DST-2 or DST-3. In this Feature #Dexample, RT may be further scaled by a constant and rounded to thenearest integer. Example values of the constant are: 64·√{square rootover (N)}, 128·√{square root over (N)}, 256·√{square root over (N)}. Inthis Feature #D example, RT may be further scaled by a constant, roundedto the nearest integer and further adjusted by an offset. Example valuesof the offset include: +1/−1, +2/−2.

Feature #E is that, in one embodiment, T₀, T₁, T₂ . . . are transformsfrom the DCT/DST families, of which examples include: DST-7, DCT-8,DST-7 with inverse sign for selected rows/columns, DCT-8 with inversesign for selected rows/columns, DST-4, or DCT-4. In one example ofFeature #E, T₀, T₁, T₂ . . . may be further scaled by a constant androunded to the nearest integer. Example values of the constant are:64·√{square root over (N)}, 128·√{square root over (N)}, 256·√{squareroot over (N)}. In one example of Feature #E, T₀, T₁, T₂ . . . may befurther scaled by a constant, rounded to the nearest integer and furtheradjusted by an offset. Example values of an offset include: +1/−1,+2/−2.

Feature #F, in one embodiment, T₀, T₁, T₂ . . . is a KLT. Feature #G, inone embodiment, N can be 4, 8, 16, 32 and/or 64. Feature #G, in oneembodiment, T₀, T₁, T₂ . . . is scaled by a pre-defined constant sbefore being embedded into R. In one example of Feature #G, the constants is √{square root over (2)}.

In Feature #I, the new resulting transform core RT may be used toreplace the DCT-2 core. In the Feature #I example, as an example, only32-point DCT-2 core and/or 64-point DCT-2 core may be replaced.

In Feature #J, whether the new resulting transform core RT is appliedcan be indicated in a VPS, SPS, PPS, or slice header, or a CTU-levelflag. In one example of Feature #J, if the new resulting transform coreRT is not applied, then the original 2N-point transform core R may beused. In one example of Feature #J, for coding a videosequence/picture/slice/CTU, if an additional transform type other thanDCT-2, e.g., DST-7/DCT-8/DCT-4/DST-4, is used, the new resultingtransform core RT may replace DCT-2 accordingly. Otherwise, if onlyDCT-2 is used, the new resulting transform core RT is not used.

Instead of embedding orthonormal transforms into a larger orthonormaltransform K, it is proposed to keep the larger orthonormal transform Kunchanged, but extract an orthonormal transform T from this largerorthonormal transform, i.e., the transform core T is a part of K. In oneembodiment, K is a 2N-point transform core, T is an N-point transformcore, and T is extracted from one half of the basis rows of RT which areanti-symmetry. In one embodiment, K is a transform from the DCT/DSTfamilies, of which examples are: DCT-2, DCT-3, DST-2 or DST-3.

In one embodiment, T is extracted from K for only certain transformsizes, e.g., 16-point, and/or 32-point. In one embodiment, DCT-4 isextracted from DCT-2, and DCT-4 is replacing the DCT-8 in AMT forcertain block sizes, e.g., 16-point, and/or 32-point. In one embodiment,a new transform core is derived by flipping DCT-4 transform bases leftand right, and this new transform core is replacing DST-7 in AMT forcertain block sizes, e.g., 16-point, 32-point.

In one embodiment, DCT-4 is extracted from DCT-2, and DCT-4 is includedas an additional transform type in AMT for certain block sizes, e.g.,4-point, 8-point, and whether DCT-4 is used depends on the block size,intra prediction mode, horizontal/vertical transform, AMT flag, AMTindex. In one embodiment, a new transform core is derived by flippingDCT-4 transform bases left and right, and this new transform core isincluded as an additional transform type in AMT for certain block sizes,e.g., 4-point, 8-point, whether this new transform core is used dependson the block size, intra prediction mode, horizontal/vertical transform,AMT flag, AMT index.

For these reasons, it is proposed to use COT cores instead oftraditional transform cores (such as DCT-2 and DST-7) in video and/orimage compression, such as transform image or residual blocks. A COTcore may be generated by doing the Kronecker product between multipleorthogonal transform cores and put them together as one larger matrixwhich is also an orthogonal transform core, and use it as the COT core.

In one embodiment, for example, given a set of N orthonormal transforms,with transform cores being denoted by T₀, T₁, . . . T_(N-1), and anadditional orthonormal transform with transform core being denoted by P,if P is divided into N matrices P₀, P₁, . . . P_(N-1), that is, if putP₀, P₁, . . . P_(N-1) together, we will get P. A new orthonormaltransform X can then be derived by putting T_(i)⊗P_(i) together as onematrix, where i=0, 1, . . . , N−1, and ⊗ denotes the Kronecker product,and use this matrix as the COT core.

In one embodiment, P is a Hadamard transform. In one embodiment, P ishorizontally divided into N matrices P₀, P₁, . . . P_(N-1) In oneembodiment, P is vertically divided into N matrices P₀, P₁, . . .P_(N-1) In one embodiment, the set of N orthonormal transform includesDCT-1˜DCT-8, DST-1˜DST-7.

In one example, N=2, and the set of N orthonormal transform are{T₀=n-point DCT-2, T₁=n-point DST-8}, P is a 2×2 Hadamard transformmatrix

$\quad{\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix},}$and P is vertically divided into P0=[1 1] and P1=[1 −1]. Aftercalculating T₀⊗P₀ and T₁⊗P₁ and putting them together, we get a new2n-point orthonormal transform X.

In one embodiment, after putting T_(i)⊗P_(i) together as one matrix, therows and/or columns are further re-ordered, which does not change theorthogonality of the new transform.

In one embodiment, before doing the Kronecker product, i.e.,T_(i)⊗P_(i), the rows and/or columns of T_(i) and/or P_(i) are furtherre-ordered, which does not change the orthogonality of the newtransform.

Referring to FIG. 5, in a first, simple example, a method of utilizing aCompound Orthonormal Transform may include the following stepsillustrated in FIG. 5. Step 500: Generate a new multi-transform datastructure by embedding one or more non-recursive transforms into alarger recursive transform. Step 501: Receive information regarding atarget data block for encoding or decoding, the target data block forencoding or decoding being one of: a compressed video or image datablock or an uncompressed video or image data block. The method of FIG. 5may also include a Step 502 that includes determining whether to use arecursive transform for the encoding or decoding of the target datablock. Similarly, Step 502 may include determining, for the encoding ordecoding of the target data block, whether to use a recursive transformor a non-recursive transform;

As shown in FIG. 5, if the Step 502 determination is YES (i.e., use therecursive transform), Step 503 is performed. Step 503 includes, when aresult of the determination is to use the recursive transform: generatethe recursive transform using the multi-transform data structure andcause or transmit information that causes the target data block to beencoded or decoded using the generated recursive transform. When thedetermination in Step 502 is NO (i.e., do not use the recursivetransforom, or alternatively use the non-recursive transform), Step 504is performed. Step 504 may include, when a result of the determinationis to not use the recursive transform, rather use one of the one or morenon-recursive transforms embedded in the multi-transform data structure.In particular, Step 504 may include causing or transmitting informationthat causes the target data block to be encoded or decoded using one ofthe one or more non-recursive transforms embedded in the multi-transformdata structure. The method of FIG. 5 may further include that at leastpart of the non-recursive transform of the multi-transform datastructure is embedded in at least part of one half of the recursivetransform of the multi-transform data structure.

The method illustrated in FIG. 5 may also include processing Steps500-504 being performed by one or more processors. The methodillustrated in FIG. 5 may also include storing, by the one or moreprocessors in a memory, the multi-transform data structure in advancebefore or during the performing of the Steps 501-504.

The method illustrated in FIG. 5 may also include that the size of thenon-recursive transform is M, where M=N/2, wherein N is the size of therecursive transform of the multi-transform data structure. The method ofFIG. 5 may further include that the recursive transform is one of:DCT-2, Hadamard transform, DCT-1, DST-1, or DST-2. The method of FIG. 5may further include that the non-recursive transform is one of: DST-7 orDCT-8.

The method of FIG. 5 may further include that the recursive transform isDCT-2 and the non-recursive transform is DST-7 or DCT-8. The method ofFIG. 5 may further include that the recursive transform is DCT-2, andthe method further comprises, when using the recursive DCT-2 transformof the multi-transform data structure: (i) determining the left half ofthe DCT-2 transform matrix based on the multi-transform data structure,and (ii) deriving the right half of the DCT-2 transform matrix accordingto the symmetry/anti-symmetry property of DCT-2 core in view of the lefthalf of the DCT-2 transform matrix.

The method of FIG. 5 may further include creating a resulting basis thatcorresponds to the original odd bases of 2N-point DCT-2 by: (i)replacing the first half of odd bases of 2N-point DCT-2 with N-pointscaled DST-7 or DCT-8 bases, and (ii) filling the second half of the oddbases of the 2N-point DCT-2 with the flipped DST-7 or DCT-8 bases plusinverse signs. The method of FIG. 5 may further include that thetransform matrices are quantized with 10-bit representation.

The method of FIG. 5 may further include deriving the multi-transformdata structure as a new orthonormal transform, which is a CompoundOrthonormal Transform (COT), by embedding an N-point non-recursivetransform into a 2N-point recursive transform, which results in the neworthonormal transform which is still an orthonormal transform due to thesymmetric/anti-symmetric characteristics of the recursive transform. Themethod of FIG. 5 may further include deriving the multi-transform datastructure as a new orthonormal transform, which is a CompoundOrthonormal Transform (COT), by embedding an N-point DCT-8 or DST-7 intoa 2N-point DCT-2, which results in the new orthonormal transform whichis still an orthonormal transform due to the symmetric/anti-symmetriccharacteristics of DCT-2.

In one example, as the multi-transform data structure, a 64-point COTcore is proposed, as shown below:

{

{aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,},

{bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw,bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, −ck, −cj, −ci,−ch, −cg, −cf, −ce, −cd, −cc, −cb, −ca, −bz, −by, −bx, −bw, −by, −bu,−bt, −bs, −br, −bq, −bp, −bo, −bn, −bm, −bl, −bk, −bj, −bi, −bh, −bg,−bf,},

{ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, −be,−bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as, −ar, −aq,−ap, −ap, −aq, −ar, −as, −at, −au, −av, −aw, −ax, −ay, −az, −ba, −bb,−bc, −bd, −be, be, bd, bc, bb, ba, az, ay, ax, aw, av, au, at, as, ar,aq, ap,},

{bg, bj, bm, bp, bs, by, by, cb, ce, ch, ck, −cj, −cg, −cd, −ca, −bx,−bu, −br, −bo, −bl, −bi, −bf, −bh, −bk, −bn, −bq, −bt, −bw, −bz, −cc,−cf, −ci, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh, bf, bi, bl, bo, br,bu, bx, ca, cd, cg, cj, −ck, −ch, −ce, −cb, −by, −by, −bs, −bp, −bm,−bj, −bg,},

{ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah,−ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai, ah,ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah,−ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai,ah,},

{bh, bm, br, bw, cb, cg, 0, −cg, −cb, −bw, −br, −bm, −bh, −bh, −bm, −br,−bw, −cb, −cg, 0, cg, cb, bw, br, bm, bh, bh, bm, br, bw, cb, cg, −cg,−cb, −bw, −br, −bm, −bh, −bh, −bm, −br, −bw, −cb, −cg, 0, cg, cb, bw,br, bm, bh, bh, bm, br, bw, cb, cg, 0, −cg, −cb, −bw, −br, −bm, −bh,},

{aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az,−bc, bc, az, aw, at, aq, aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq,−aq, −at, −aw, −az, −bc, 0, bc, az, aw, at, aq, aq, at, aw, az, bc, −bc,−az, −aw, −at, −aq, −aq, −at, −aw, −az, −bc, 0, bc, az, aw, at, aq,},

{bi, bp, bw, cd, ck, −cf, −by, −br, −bk, −bg, −bn, −bu, −cb, −ci, ch,ca, bt, bm, bf, bl, bs, bz, cg, −cj, −cc, −by, −bo, −bh, −bj, −bq, −bx,−ce, ce, bx, bq, bj, bh, bo, by, cc, cj, −cg, −bz, −bs, −bl, −bf, −bm,−bt, −ca, −ch, ci, cb, bu, bn, bg, bk, br, by, cf, −ck, −cd, −bw, −bp,−bi,},

{ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae,ad,},

{bj, bs, cb, ck, −cd, −bu, −bl, −bh, −bq, −bz, −ci, cf, bw, bn, bf, bo,bx, cg, −ch, −by, −bp, −bg, −bm, −by, −ce, cj, ca, br, bi, bk, bt, cc,−cc, −bt, −bk, −bi, −br, −ca, −cj, ce, by, bm, bg, bp, by, ch, −cg, −bx,−bo, −bf, −bn, −bw, −cf, ci, bz, bq, bh, bl, bu, cd, −ck, −cb, −bs,−bj,},

{ar, aw, bb, −be, −az, −au, −ap, −at, −ay, −bd, bc, ax, as, aq, ay, ba,−ba, −ay, −aq, −as, −ax, −bc, bd, ay, at, ap, au, az, be, −bb, −aw, −ar,−ar, −aw, −bb, be, az, au, ap, at, ay, bd, −bc, −ax, −as, −aq, −ay, −ba,ba, ay, aq, as, ax, bc, −bd, −ay, −at, −ap, −au, −az, −be, bb, aw, ar,},

{bk, by, cg, −cf, −bu, −bj, −bl, −bw, −ch, ce, bt, bi, bm, bx, ci, −cd,−bs, −bh, −bn, −by, −cj, cc, br, bg, bo, bz, ck, −cb, −bq, −bf, −bp,−ca, ca, bp, bf, bq, cb, −ck, −bz, −bo, −bg, −br, −cc, cj, by, bn, bh,bs, cd, −ci, −bx, −bm, −bi, −bt, −ce, ch, bw, bl, bj, bu, cf, −cg, −by,−bk,},

{ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai,−ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al, ai,ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai,−ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al,ai,},

{bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl,−by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, −by,−bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl, −bl, −by, 0, by, bl, bl,by, 0, −by, −bl, −bl, −by, 0, by, bl, bl, by, 0, −by, −bl,},

{as, az, −be, −ax, −aq, −au, −bb, bc, av, ap, aw, bd, −ba, −at, −ar,−ay, ay, ar, at, ba, −bd, −aw, −ap, −av, −bc, bb, au, aq, ax, be, −az,−as, −as, −az, be, ax, aq, au, bb, −bc, −av, −ap, −aw, −bd, ba, at, ar,ay, −ay, −ar, −at, −ba, bd, aw, ap, av, bc, −bb, −au, −aq, −ax, −be, az,as,},

{bm, cb, −cg, −br, −bh, −bw, 0, bw, bh, br, cg, −cb, −bm, −bm, −cb, cg,br, bh, bw, 0, −bw, −bh, −br, −cg, cb, bm, bm, cb, −cg, −br, −bh, −bw,bw, bh, br, cg, −cb, −bm, −bm, −cb, cg, br, bh, bw, 0, −bw, −bh, −br,−cg, cb, bm, bm, cb, −cg, −br, −bh, −bw, 0, bw, bh, br, cg, −cb, −bm,},

{ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac,ab,},

{bn, ce, −cb, −bk, −bq, −ch, by, bh, bt, ck, −by, −bf, −bw, cj, bs, bi,bz, −cg, −bp, −bl, −cc, cd, bm, bo, cf, −ca, −bj, −br, −ci, bx, bg, bu,−bu, −bg, −bx, ci, br, bj, ca, −cf, −bo, −bm, −cd, cc, bl, bp, cg, −bz,−bi, −bs, −cj, bw, bf, by, −ck, −bt, −bh, −by, ch, bq, bk, cb, −ce,−bn,},

{at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw,−aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at,−at, −bc, az, aq, aw, 0, −aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw,aw, aq, az, −bc, −at, −at, −bc, az, aq, aw, 0, −aw, −aq, −az, bc, at,},

{bo, ch, −bw, −bg, −bz, ce, bl, br, ck, −bt, −bj, −cc, cb, bi, bu, −cj,−bq, −bm, −cf, by, bf, bx, −cg, −bn, −bp, −ci, by, bh, ca, −cd, −bk,−bs, bs, bk, cd, −ca, −bh, −by, ci, bp, bn, cg, −bx, −bf, −by, cf, bm,bq, cj, −bu, −bi, −cb, cc, bj, bt, −ck, −br, −bl, −ce, bz, bg, bw, −ch,−bo,},

{aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj,−aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao, aj,aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj,−aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao,aj,},

{bp, ck, −br, −bn, −ci, bt, bl, cg, −by, −bj, −ce, bx, bh, cc, −bz, −bf,−ca, cb, bg, by, −cd, −bi, −bw, cf, bk, bu, −ch, −bm, −bs, cj, bo, bq,−bq, −bo, −cj, bs, bm, ch, −bu, −bk, −cf, bw, bi, cd, −by, −bg, −cb, ca,bf, bz, −cc, −bh, −bx, ce, bj, by, −cg, −bl, −bt, ci, bn, br, −ck,−bp,},

{au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, au,au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au,au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, −au, −au, 0, au, au,0, −au, −au, 0, au, au, 0, −au, −au, 0, au,}, {bq, −cj, −bm, −bu, cf,bi, by, −cb, −bf, −cc, bx, bj, cg, −bt, −bn, −ck, bp, br, −ci, −bl, −by,ce, bh, bz, −ca, −bg, −cd, bw, bk, ch, −bs, −bo, bo, bs, −ch, −bk, −bw,cd, bg, ca, −bz, −bh, −ce, by, bl, ci, −br, −bp, ck, bn, bt, −cg, −bj,−bx, cc, bf, cb, −by, −bi, −cf, bu, bm, cj, −bq,}, {ae, −ag, −ad, −af,af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af,af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af,af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae, ae, −ag, −ad, −af,af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae,},

{br, −cg, −bh, −cb, bw, bm, 0, −bm, −bw, cb, bh, cg, −br, −br, cg, bh,cb, −bw, −bm, 0, bm, bw, −cb, −bh, −cg, br, br, −cg, −bh, −cb, bw, bm,−bm, −bw, cb, bh, cg, −br, −br, cg, bh, cb, −bw, −bm, 0, bm, bw, −cb,−bh, −cg, br, br, −cg, −bh, −cb, bw, bm, 0, −bm, −bw, cb, bh, cg, −br,},

{av, −bc, −ap, −bb, aw, au, −bd, −aq, −ba, ax, at, −be, −ar, −az, ay,as, −as, −ay, az, ar, be, −at, −ax, ba, aq, bd, −au, −aw, bb, ap, bc,−ay, −ay, bc, ap, bb, −aw, −au, bd, aq, ba, −ax, −at, be, ar, az, −ay,−as, as, ay, −az, −ar, −be, at, ax, −ba, −aq, −bd, au, aw, −bb, −ap,−bc, ay,},

{bs, −cd, −bh, −ci, bn, bx, −by, −bm, cj, bi, cc, −bt, −br, ce, bg, ch,−bo, −bw, bz, bl, −ck, −bj, −cb, bu, bq, −cf, −bf, −cg, bp, by, −ca,−bk, bk, ca, −by, −bp, cg, bf, cf, −bq, −bu, cb, bj, ck, −bl, −bz, bw,bo, −ch, −bg, −ce, br, bt, −cc, −bi, −cj, bm, by, −bx, −bn, ci, bh, cd,−bs,},

{ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak,−ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am, ak,ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak,−ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am,ak,},

{bt, −ca, −bm, ch, bf, ci, −bl, −cb, bs, bu, −bz, −bn, cg, bg, cj, −bk,−cc, br, by, −by, −bo, cf, bh, ck, −bj, −cd, bq, bw, −bx, −bp, ce, bi,−bi, −ce, bp, bx, −bw, −bq, cd, bj, −ck, −bh, −cf, bo, by, −by, −br, cc,bk, −cj, −bg, −cg, bn, bz, −bu, −bs, cb, bl, −ci, −bf, −ch, bm, ca,−bt,},

{aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq,aq, bc, −at, −az, aw, aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw,−aw, az, at, −bc, −aq, 0, aq, bc, −at, −az, aw, aw, −az, −at, bc, aq,−aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq, 0, aq, bc, −at, −az, aw,},

{bu, −bx, −br, ca, bo, −cd, −bl, cg, bi, −cj, −bf, −ck, bh, ch, −bk,−ce, bn, cb, −bq, −by, bt, by, −bw, −bs, bz, bp, −cc, −bm, cf, bj, −ci,−bg, bg, ci, −bj, −cf, bm, cc, −bp, −bz, bs, bw, −by, −bt, by, bq, −cb,−bn, ce, bk, −ch, −bh, ck, bf, cj, −bi, −cg, bl, cd, −bo, −ca, br, bx,−bu,},

{aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa,aa,},

{by, −bu, −bw, bt, bx, −bs, −by, br, bz, −bq, −ca, bp, cb, −bo, −cc, bn,cd, −bm, −ce, bl, cf, −bk, −cg, bj, ch, −bi, −ci, bh, cj, −bg, −ck, bf,−bf, ck, bg, −cj, −bh, ci, bi, −ch, −bj, cg, bk, −cf, −bl, ce, bm, −cd,−bn, cc, bo, −cb, −bp, ca, bq, −bz, −br, by, bs, −bx, −bt, bw, bu,−by,},

{ax, −aw, −ay, av, az, −au, −ba, at, bb, −as, −bc, ar, bd, −aq, −be, ap,−ap, be, aq, −bd, −ar, bc, as, −bb, −at, ba, au, −az, −ay, ay, aw, −ax,−ax, aw, ay, −ay, −az, au, ba, −at, −bb, as, bc, −ar, −bd, aq, be, −ap,ap, −be, −aq, bd, ar, −bc, −as, bb, at, −ba, −au, az, av, −ay, −aw,ax,},

{bw, −br, −cb, bm, cg, −bh, 0, bh, −cg, −bm, cb, br, −bw, −bw, br, cb,−bm, −cg, bh, 0, −bh, cg, bm, −cb, −br, bw, bw, −br, −cb, bm, cg, −bh,bh, −cg, −bm, cb, br, −bw, −bw, br, cb, −bm, −cg, bh, 0, −bh, cg, bm,−cb, −br, bw, bw, −br, −cb, bm, cg, −bh, 0, bh, −cg, −bm, cb, br, −bw,},

{al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al,−al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj, al,al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al,−al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj,al,},

{bx, −bo, −cg, bf, −ch, −bn, by, bw, −bp, −cf, bg, −ci, −bm, bz, by,−bq, −ce, bh, −cj, −bl, ca, bu, −br, −cd, bi, −ck, −bk, cb, bt, −bs,−cc, bj, −bj, cc, bs, −bt, −cb, bk, ck, −bi, cd, br, −bu, −ca, bl, cj,−bh, ce, bq, −by, −bz, bm, ci, −bg, cf, bp, −bw, −by, bn, ch, −bf, cg,bo, −bx,},

{ay, −at, −bd, ap, −bc, −au, ax, az, −as, −be, aq, −bb, −av, aw, ba,−ar, ar, −ba, −aw, av, bb, −aq, be, as, −az, −ax, au, bc, −ap, bd, at,−ay, −ay, at, bd, −ap, bc, au, −ax, −az, as, be, −aq, bb, av, −aw, −ba,ar, −ar, ba, aw, −av, −bb, aq, −be, −as, az, ax, −au, −bc, ap, −bd, −at,ay,},

{by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl,0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl, bl, −by,−by, bl, 0, −bl, by, by, −bl, 0, bl, −by, −by, bl, 0, −bl, by, by, −bl,0, bl, −by, −by, bl, 0, −bl, by, by, −bl, 0, bl, −by,},

{af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad,af,},

{bz, −bi, cg, bs, −bp, −cj, bl, −bw, −cc, bf, −cd, −by, bm, −ck, −bo,bt, cf, −bh, ca, by, −bj, ch, br, −bq, −ci, bk, −bx, −cb, bg, −ce, −bu,bn, −bn, bu, ce, −bg, cb, bx, −bk, ci, bq, −br, −ch, bj, −by, −ca, bh,−cf, −bt, bo, ck, −bm, by, cd, −bf, cc, bw, −bl, cj, bp, −bs, −cg, bi,−bz,},

{az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at,−at, aw, bc, −aq, az, az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az,−az, aq, −bc, −aw, at, 0, −at, aw, bc, −aq, az, az, −aq, bc, aw, −at,at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at, 0, −at, aw, bc, −aq, az,},

{ca, −bf, cb, bz, −bg, cc, by, −bh, cd, bx, −bi, ce, bw, −bj, cf, by,−bk, cg, bu, −bl, ch, bt, −bm, ci, bs, −bn, cj, br, −bo, ck, bq, −bp,bp, −bq, −ck, bo, −br, −cj, bn, −bs, −ci, bm, −bt, −ch, bl, −bu, −cg,bk, −by, −cf, bj, −bw, −ce, bi, −bx, −cd, bh, −by, −cc, bg, −bz, −cb,bf, −ca,},

{am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am,−am, ah, −al, −an, ai, −ao, aj, −aj, ao, ak, −ai, an, al, −ah, am, am,−ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am, −am,ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai, an, al, −ah, am,},

{cb, −bh, bw, cg, −bm, br, 0, −br, bm, −cg, −bw, bh, −cb, −cb, bh, −bw,−cg, bm, −br, 0, br, −bm, cg, bw, −bh, cb, cb, −bh, bw, cg, −bm, br,−br, bm, −cg, −bw, bh, −cb, −cb, bh, −bw, −cg, bm, −br, 0, br, −bm, cg,bw, −bh, cb, cb, −bh, bw, cg, −bm, br, 0, −br, bm, −cg, −bw, bh, −cb,},

{ba, −aq, ax, bd, −at, au, −be, −aw, ar, −bb, −az, ap, −ay, −bc, as,−av, av, −as, bc, ay, −ap, az, bb, −ar, aw, be, −au, at, −bd, −ax, aq,−ba, −ba, aq, −ax, −bd, at, −au, be, aw, −ar, bb, az, −ap, ay, bc, −as,av, −av, as, −bc, −ay, ap, −az, −bb, ar, −aw, −be, au, −at, bd, ax, −aq,ba,},

{cc, −bk, br, −cj, −by, bg, −by, −cg, bo, −bn, cf, bz, −bh, bu, ck, −bs,bj, −cb, −cd, bl, −bq, ci, bw, −bf, bx, ch, −bp, bm, −ce, −ca, bi, −bt,bt, −bi, ca, ce, −bm, bp, −ch, −bx, bf, −bw, −ci, bq, −bl, cd, cb, −bj,bs, −ck, −bu, bh, −bz, −cf, bn, −bo, cg, by, −bg, by, cj, −br, bk,−cc,},

{ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab,ac,},

{cd, −bn, bm, −cc, −ce, bo, −bl, cb, cf, −bp, bk, −ca, −cg, bq, −bj, bz,ch, −br, bi, −by, −ci, bs, −bh, bx, cj, −bt, bg, −bw, −ck, bu, −bf, by,−by, bf, −bu, ck, bw, −bg, bt, −cj, −bx, bh, −bs, ci, by, −bi, br, −ch,−bz, bj, −bq, cg, ca, −bk, bp, −cf, −cb, bl, −bo, ce, cc, −bm, bn,−cd,},

{bb, −at, as, −ba, −bc, au, −ar, az, bd, −av, aq, −ay, −be, aw, −ap, ax,−ax, ap, −aw, be, ay, −aq, av, −bd, −az, ar, −au, bc, ba, −as, at, −bb,−bb, at, −as, ba, bc, −au, ar, −az, −bd, av, −aq, ay, be, −aw, ap, −ax,ax, −ap, aw, −be, −ay, aq, −av, bd, az, −ar, au, −bc, −ba, as, −at,bb,},

{ce, −bq, bh, −by, cj, bz, −bl, bm, −ca, −ci, bu, −bg, br, −cf, −cd, bp,−bi, bw, −ck, −by, bk, −bn, cb, ch, −bt, bf, −bs, cg, cc, −bo, bj, −bx,bx, −bj, bo, −cc, −cg, bs, −bf, bt, −ch, −cb, bn, −bk, by, ck, −bw, bi,−bp, cd, cf, −br, bg, −bu, ci, ca, −bm, bl, −bz, −cj, by, −bh, bq,−ce,},

{an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an,−an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak, an,an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an,−an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak,an,},

{cf, −bt, bh, −bo, ca, ck, −by, bm, −bj, by, −ch, −cd, br, −bf, bq, −cc,−ci, bw, −bk, bl, −bx, cj, cb, −bp, bg, −bs, ce, cg, −bu, bi, −bn, bz,−bz, bn, −bi, bu, −cg, −ce, bs, −bg, bp, −cb, −cj, bx, −bl, bk, −bw, ci,cc, −bq, bf, −br, cd, ch, −by, bj, −bm, by, −ck, −ca, bo, −bh, bt,−cf,},

{bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az,az, −at, aq, −aw, bc, bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc,−bc, aw, −aq, at, −az, 0, az, −at, aq, −aw, bc, bc, −aw, aq, −at, az,−az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az, 0, az, −at, aq, −aw, bc,},

{cg, −bw, bm, −bh, br, −cb, 0, cb, −br, bh, −bm, bw, −cg, −cg, bw, −bm,bh, −br, cb, 0, −cb, br, −bh, bm, −bw, cg, cg, −bw, bm, −bh, br, −cb,cb, −br, bh, −bm, bw, −cg, −cg, bw, −bm, bh, −br, cb, 0, −cb, br, −bh,bm, −bw, cg, cg, −bw, bm, −bh, br, −cb, 0, cb, −br, bh, −bm, bw, −cg,},

{ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af,ag,},

{ch, −bz, br, −bj, bi, −bq, by, −cg, −ci, ca, −bs, bk, −bh, bp, −bx, cf,cj, −cb, bt, −bl, bg, −bo, bw, −ce, −ck, cc, −bu, bm, −bf, bn, −by, cd,−cd, by, −bn, bf, −bm, bu, −cc, ck, ce, −bw, bo, −bg, bl, −bt, cb, −cj,−cf, bx, −bp, bh, −bk, bs, −ca, ci, cg, −by, bq, −bi, bj, −br, bz,−ch,},

{bd, −az, av, −ar, aq, −au, ay, −bc, −be, ba, −aw, as, −ap, at, −ax, bb,−bb, ax, −at, ap, −as, aw, −ba, be, bc, −ay, au, −aq, ar, −av, az, −bd,−bd, az, −av, ar, −aq, au, −ay, bc, be, −ba, aw, −as, ap, −at, ax, −bb,bb, −ax, at, −ap, as, −aw, ba, −be, −bc, ay, −au, aq, −ar, av, −az,bd,},

{ci, −cc, bw, −bq, bk, −bf, bl, −br, bx, −cd, cj, ch, −cb, by, −bp, bj,−bg, bm, −bs, by, −ce, ck, cg, −ca, bu, −bo, bi, −bh, bn, −bt, bz, −cf,cf, −bz, bt, −bn, bh, −bi, bo, −bu, ca, −cg, −ck, ce, −by, bs, −bm, bg,−bj, bp, −by, cb, −ch, −cj, cd, −bx, br, −bl, bf, −bk, bq, −bw, cc,−ci,},

{ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao,−ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an, ao,ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao,−ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an,ao,},

{cj, −cf, cb, −bx, bt, −bp, bl, −bh, bg, −bk, bo, −bs, bw, −ca, ce, −ci,−ck, cg, −cc, by, −bu, bq, −bm, bi, −bf, bj, −bn, br, −by, bz, −cd, ch,−ch, cd, −bz, by, −br, bn, −bj, bf, −bi, bm, −bq, bu, −by, cc, −cg, ck,ci, −ce, ca, −bw, bs, −bo, bk, −bg, bh, −bl, bp, −bt, bx, −cb, cf,−cj,},

{be, −bc, ba, −ay, aw, −au, as, −aq, ap, −ar, at, −ay, ax, −az, bb, −bd,bd, −bb, az, −ax, ay, −at, ar, −ap, aq, −as, au, −aw, ay, −ba, bc, −be,−be, bc, −ba, ay, −aw, au, −as, aq, −ap, ar, −at, ay, −ax, az, −bb, bd,−bd, bb, −az, ax, −ay, at, −ar, ap, −aq, as, −au, aw, −ay, ba, −bc,be,},

{ck, −ci, cg, −ce, cc, −ca, by, −bw, bu, −bs, bq, −bo, bm, −bk, bi, −bg,bf, −bh, bj, −bl, bn, −bp, br, −bt, by, −bx, bz, −cb, cd, −cf, ch, −cj,cj, −ch, cf, −cd, cb, −bz, bx, −by, bt, −br, bp, −bn, bl, −bj, bh, −bf,bg, −bi, bk, −bm, bo, −bq, bs, −bu, bw, −by, ca, −cc, ce, −cg, ci,−ck,},

}

where {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh,bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz,ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl} are pre-defined numbers.

An example value of the above pre-defined numbers (elements in the64−point COT core) are {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al,am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd,be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by,bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, cl}={64, 83,36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 89, 88, 87, 84, 81,77, 73, 67, 62, 55, 48, 41, 33, 25, 17, 8, 90, 90, 89, 89, 88, 87, 85,84, 82, 81, 78, 76, 74, 71, 69, 66, 63, 60, 56, 53, 49, 46, 42, 38, 34,30, 26, 21, 17, 13, 9, 4}.

An example value of the elements in 64-point COT core is: {aa, ab, ac,ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au,av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj, bk, bl, bm,bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz, ca, cb, cc, cd, ce,cf, cg, ch, ci, cj, ck}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57,43, 25, 9, 90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17,9, 90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63,60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4}

From the above transform core, the 2-point DCT-2 can be extracted, asshown below:

aa aa aa −aa 

From the above transform core, the 4-point DCT-2 can be extracted, asshown below:

aa  aa  aa  aa ab  ac −ac −ab aa −aa −aa  aa ac −ab  ab −ac

From the above transform core, the 8-point DCT-2 can be extracted, asshown below:

aa aa aa aa aa aa aa aa ad ae af ag -ag -af -ae -ad ab ac -ac -ab -ab-ac ac ab ae -ag -ad -af af ad ag -ae aa -aa -aa aa aa -aa -aa aa af -adag ae -ae -ag ad -af ac -ab ab -ac -ac ab -ab ac ag -af ae -ad ad -ae af-ag

From the above transform core, the 16-point DCT-2 can be extracted, asshown below:

aa aa aa aa aa aa aa aa aa aa aa aa aa aa aa aa ah ar aj ak al am an ao-ao -an -am -al -ak -aj -ar -ah ad ae af ag -ag -af -ae -ad -ad -ae -af-ag ag af ae ad ar al ao -am -aj -ah -ak -an an ak ah aj am -ao -al -arab ac -ac -ab -ab -ac ac ab ab ac -ac -ab -ab -ac ac ab aj ao -ak -ar-an al ah am -am -ah -al an ar ak -ao -aj ae -ag -ad -af af ad ag -ae-ae ag ad af -af -ad -ag ae ak -am -ar ao ah an -aj -al al aj -an -ah-ao ar am -ak aa -aa -aa aa aa -aa -aa aa aa -aa -aa aa aa -aa -aa aa al-aj -an ah -ao -ar am ak -ak -am ar ao -ah an aj -al af -ad ag ae -ae-ag ad -af -af ad -ag -ae ae ag -ad af am -ah al an -ar ak ao -aj aj -ao-ak ar -an -al ah -am ac -ab ab -ac -ac ab -ab ac ac -ab ab -ac -ac ab-ab ac an -ak ah -aj am ao -al ar -ar al -ao -am aj -ah ak -an ag -af ae-ad ad -ae af -ag -ag af -ae ad -ad ae -af ag ao -an am -al ak -aj ar-ah ah -ar aj -ak al -am an -ao

From the above transform core, the 32-point COT which replaces 32-pointDCT-2 can be extracted, as shown below:

{

{aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,},

{ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, −be,−bd, −bc, −bb, −ba, −az, −ay, −ax, −aw, −av, −au, −at, −as, −ar, −aq,−ap,},

{ah, ai, aj, ak, al, am, an, ao, −ao, −an, −am, −al, −ak, −aj, −ai, −ah,−ah, −ai, −aj, −ak, −al, −am, −an, −ao, ao, an, am, al, ak, aj, ai,ah,},

{aq, at, aw, az, bc, 0, −bc, −az, −aw, −at, −aq, −aq, −at, −aw, −az,−bc, bc, az, aw, at, aq, aq, at, aw, az, bc, 0, −bc, −az, −aw, −at,−aq,},

{ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae, ad,ad, ae, af, ag, −ag, −af, −ae, −ad, −ad, −ae, −af, −ag, ag, af, ae,ad,},

{ar, aw, bb, −be, −az, −au, −ap, −at, −ay, −bd, bc, ax, as, aq, av, ba,−ba, −av, −aq, −as, −ax, −bc, bd, ay, at, ap, au, az, be, −bb, −aw,−ar,},

{ai, al, ao, −am, −aj, −ah, −ak, −an, an, ak, ah, aj, am, −ao, −al, −ai,−ai, −al, −ao, am, aj, ah, ak, an, −an, −ak, −ah, −aj, −am, ao, al,ai,},

{as, az, −be, −ax, −aq, −au, −bb, bc, av, ap, aw, bd, −ba, −at, −ar,−ay, ay, ar, at, ba, −bd, −aw, −ap, −av, −bc, bb, au, aq, ax, be, −az,−as,},

{ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac, ab,ab, ac, −ac, −ab, −ab, −ac, ac, ab, ab, ac, −ac, −ab, −ab, −ac, ac,ab,},

{at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at, −at, −bc, az, aq, aw,−aw, −aq, −az, bc, at, at, bc, −az, −aq, −aw, 0, aw, aq, az, −bc, −at,},

{aj, ao, −ak, −ai, −an, al, ah, am, −am, −ah, −al, an, ai, ak, −ao, −aj,−aj, −ao, ak, ai, an, −al, −ah, −am, am, ah, al, −an, −ai, −ak, ao,aj,},

{au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au, −au, au,au, 0, −au, −au, 0, au, au, 0, −au, −au, 0, au, au, 0, −au,},

{ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag, ae,ae, −ag, −ad, −af, af, ad, ag, −ae, −ae, ag, ad, af, −af, −ad, −ag,ae,},

{av, −bc, −ap, −bb, aw, au, −bd, −aq, −ba, ax, at, −be, −ar, −az, ay,as, −as, −ay, az, ar, be, −at, −ax, ba, aq, bd, −au, −aw, bb, ap, bc,−av,},

{ak, −am, −ai, ao, ah, an, −aj, −al, al, aj, −an, −ah, −ao, ai, am, −ak,−ak, am, ai, −ao, −ah, −an, aj, al, −al, −aj, an, ah, ao, −ai, −am,ak,},

{aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw, −aw, az, at, −bc, −aq,aq, bc, −at, −az, aw, aw, −az, −at, bc, aq, 0, −aq, −bc, at, az, −aw,},

{aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa,aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa, aa, aa, −aa, −aa,aa,},

{ax, −aw, −ay, av, az, −au, −ba, at, bb, −as, −bc, ar, bd, −aq, −be, ap,−ap, be, aq, −bd, −ar, bc, as, −bb, −at, ba, au, −az, −av, ay, aw,−ax,},

{al, −aj, −an, ah, −ao, −ai, am, ak, −ak, −am, ai, ao, −ah, an, aj, −al,−al, aj, an, −ah, ao, ai, −am, −ak, ak, am, −ai, −ao, ah, −an, −aj,al,},

{ay, −at, −bd, ap, −bc, −au, ax, az, −as, −be, aq, −bb, −av, aw, ba,−ar, ar, −ba, −aw, av, bb, −aq, be, as, −az, −ax, au, bc, −ap, bd, at,−ay,},

{af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad, af,af, −ad, ag, ae, −ae, −ag, ad, −af, −af, ad, −ag, −ae, ae, ag, −ad,af,},

{az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az, −az, aq, −bc, −aw, at,−at, aw, bc, −aq, az, az, −aq, bc, aw, −at, 0, at, −aw, −bc, aq, −az,},

{am, −ah, al, an, −ai, ak, ao, −aj, aj, −ao, −ak, ai, −an, −al, ah, −am,−am, ah, −al, −an, ai, −ak, −ao, aj, −aj, ao, ak, −ai, an, al, −ah,am,},

{ba, −aq, ax, bd, −at, au, −be, −aw, ar, −bb, −az, ap, −ay, −bc, as,−av, av, −as, bc, ay, −ap, az, bb, −ar, aw, be, −au, at, −bd, −ax, aq,−ba,},

{ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab, ac,ac, −ab, ab, −ac, −ac, ab, −ab, ac, ac, −ab, ab, −ac, −ac, ab, −ab,ac,},

{bb, −at, as, −ba, −bc, au, −ar, az, bd, −av, aq, −ay, −be, aw, −ap, ax,−ax, ap, −aw, be, ay, −aq, av, −bd, −az, ar, −au, bc, ba, −as, at,−bb,},

{an, −ak, ah, −aj, am, ao, −al, ai, −ai, al, −ao, −am, aj, −ah, ak, −an,−an, ak, −ah, aj, −am, −ao, al, −ai, ai, −al, ao, am, −aj, ah, −ak,an,},

{bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc, −bc, aw, −aq, at, −az,az, −at, aq, −aw, bc, bc, −aw, aq, −at, az, 0, −az, at, −aq, aw, −bc,},

{ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af, ag,ag, −af, ae, −ad, ad, −ae, af, −ag, −ag, af, −ae, ad, −ad, ae, −af,ag,},

{bd, −az, av, −ar, aq, −au, ay, −bc, −be, ba, −aw, as, −ap, at, −ax, bb,−bb, ax, −at, ap, −as, aw, −ba, be, bc, −ay, au, −aq, ar, −av, az,−bd,},

{ao, −an, am, −al, ak, −aj, ai, −ah, ah, −ai, aj, −ak, al, −am, an, −ao,−ao, an, −am, al, −ak, aj, −ai, ah, −ah, ai, −aj, ak, −al, am, −an,ao,},

{be, −bc, ba, −ay, aw, −au, as, −aq, ap, −ar, at, −av, ax, −az, bb, −bd,bd, −bb, az, −ax, av, −at, ar, −ap, aq, −as, au, −aw, ay, −ba, bc,−be,},

}

An example value of the elements in 32-point COT core is: {aa, ab, ac,ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar, as, at, au,av, aw, ax, ay, az, ba, bb, bc, bd, be}={64, 83, 36, 89, 75, 50, 18, 90,87, 80, 70, 57, 43, 25, 9, 90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49,41, 33, 25, 17, 9}

In addition, the 16-point DCT-8 transform core can be also extracted asbelow, and the 16-point DST-7 transform core can be derived by flippingthe 16-point DCT-8 transform core left and right plus sign changes onthe odd rows.

ap aq ar as at au av aw ax ay az ba bb bc bd be aq at aw az bc 0 -bc -az-aw -at -aq -aq -at -aw -az -bc ar aw bb -be -az -au -ap -at -ay -bd bcax as aq av ba as az -be -ax -aq -au -bb bc av ap aw bd -ba -at -ar -ayat bc -az -aq -aw 0 aw aq az -bc -at -at -bc az aq aw au 0 -au -au 0 auau 0 -au -au 0 au au 0 -au -au av -bc -ap -bb aw au -bd -aq -ba ax at-be -ar -az ay as aw -az -at bc aq 0 -aq -bc at az -aw -aw az at -bc -aqax -aw -ay av az -au -ba at bb -as -bc ar bd -aq -be ap ay -at -bd ap-bc -au ax az -as -be aq -bb -av aw ba -ar az -aq bc aw -at 0 at -aw -bcaq -az -az aq -bc -aw at ba -aq ax bd -at au -be -aw ar -bb -az ap -ay-bc as -av bb -at as -ba -bc au -ar az bd -av aq -ay -be aw -ap ax bc-aw aq -at az 0 -az at -aq aw -bc -bc aw -aq at -az bd -az av -ar aq -auay -bc -be ba -aw as -ap at -ax bb be -bc ba -ay aw -au as -aq ap -ar at-av ax -az bb -bd

In addition, the 32-point DCT-8 transform core can be also extracted asbelow, and the 32-point DST-7 transform core can be derived by flippingthe 32-point DCT-8 transform core left and right plus sign changes onthe odd rows.

The 32-point DCT-8 transform core is shown below:

{

{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x,y, z, A, B, C, D, E, F,},

{b, e, h, k, n, q, t, w, z, C, F, −E, −B, −y, −v, −s, −p, −m, −j, −g,−d, −a, −c, −f, −i, −l, −o, −r, −u, −x, −A, −D,},

{c, h, m, r, w, B, 0, −B, −w, −r, −m, −h, −c, −c, −h, −m, −r, −w, −B, 0,B, w, r, m, h, c, c, h, in, r, w, B,},

{d, k, r, y, F, −A, −t, −m, −f, −b, −i, −p, −w, −D, C, v, o, h, a, g, n,u, B, −E, −x, −q, −j, −c, −e, −l, −s, −z,},

{e, n, w, F, −y, −p, −g, −c, −l, −u, −D, A, r, i, a, j, s, B, −C, −t,−k, −b, −h, −q, −z, E, v, m, d, f, o, x,},

{f, q, B, −A, −p, −e, −g, −r, −C, z, o, d, h, s, D, −y, −n, −c, −i, −t,−E, x, m, b, j, u, F, −w, −l, −a, −k, −v,},

{g, t, 0, −t, −g, −g, −t, 0, t, g, g, t, 0, −t, −g, −g, −t, 0, t, g, g,t, 0, −t, −g, −g, −t, 0, t, g, g, t,},

{h, w, −B, −m, −c, −r, 0, r, c, m, B, −w, −h, −h, −w, B, m, c, r, 0, −r,−c, −m, −B, w, h, h, w, −B, −m, −c, −r,},

{i, z, −w, −f, −l, −C, t, c, o, F, −q, −a, −r, E, n, d, u, −B, −k, −g,−x, y, h, j, A, −v, −e, −m, −D, s, b, p,},

{j, C, −r, −b, −u, z, g, m, F, −o, −e, −x, w, d, p, −E, −l, −h, −A, t,a, s, −B, −i, −k, −D, q, c, v, −y, −f, −n,},

{k, F, −m, −i, −D, o, g, B, −q, −e, −z, s, c, x, −u, −a, −v, w, b, t,−y, −d, −r, A, f, p, −C, −h, −n, E, j, l,}, {l, −E, −h, −p, A, d, t, −w,−a, −x, s, e, B, −o, −i, −F, k, m, −D, −g, −q, z, c, u, −v, −b, −y, r,f, C, −n, −j,},

{m, −B, −c, −w, r, h, 0, −h, −r, w, c, B, −m, −m, B, c, w, −r, −h, 0, h,r, −w, −c, −B, m, m, −B, −c, −w, r, h,},

{n, −y, −c, −D, i, s, −t, −h, E, d, x, −o, −m, z, b, C, −j, −r, u, g,−F, −e, −w, p, l, −A, −a, −B, k, q, −v, −f,},

{o, −v, −h, C, a, D, −g, −w, n, p, −u, −i, B, b, E, −f, −x, m, q, −t,−j, A, c, F, −e, −y, l, r, −s, −k, z, d,},

{p, −s, −m, v, j, −y, −g, B, d, −E, −a, −F, c, C, −f, −z, i, w, −l, −t,o, q, −r, −n, u, k, −x, −h, A, e, −D, −b,},

{q, −p, −r, o, s, −n, −t, m, u, −l, −v, k, w, −j, −x, i, y, −h, −z, g,A, −f, −B, e, C, −d, −D, c, E, −b, −F, a,},

{r, −m, −w, h, B, −c, 0, c, −B, −h, w, m, −r, −r, m, w, −h, −B, c, 0,−c, B, h, −w, −m, r, r, −m, −w, h, B, −c,},

{s, −j, −B, a, −C, −i, t, r, −k, −A, b, −D, −h, u, q, −l, −z, c, −E, −g,v, p, −m, −y, d, −F, −f, w, o, −n, −x, e,},

{t, −g, 0, g, −t, −t, g, 0, −g, t, t, −g, 0, g, −t, −t, g, 0, −g, t, t,−g, 0, g, −t, −t, g, 0, −g, t, t, −g,},

{u, −d, B, n, −k, −E, g, −r, −x, a, −y, −q, h, −F, −j, o, A, −c, v, t,−e, C, m, −l, −D, f, −s, −w, b, −z, −p, i,},

{v, −a, w, u, −b, x, t, −c, y, s, −d, z, r, −e, A, q, −f, B, p, −g, C,o, −h, D, n, −i, E, m, −j, F, l, −k,},

{w, −c, r, B, −h, m, 0, −m, h, −B, −r, c, −w, −w, c, −r, −B, h, −m, 0,m, −h, B, r, −c, w, w, −c, r, B, −h, m,},

{x, −f, m, −E, −q, b, −t, −B, j, −i, A, u, −c, p, F, −n, e, −w, −y, g,−l, D, r, −a, s, C, −k, h, −z, −v, d, −o,},

{y, −i, h, −x, −z, j, −g, w, A, −k, f, −v, −B, l, −e, u, C, −m, d, −t,−D, n, −c, s, E, −o, b, −r, −F, p, −a, q,},

{z, −l, c, −q, E, u, −g, h, −v, −D, p, −b, m, −A, −y, k, −d, r, −F, −t,f, −i, w, C, −o, a, −n, B, x, −j, e, −s,},

{A, −o, c, −j, v, F, −t, h, −e, q, −C, −y, m, −a, l, −x, −D, r, −f, g,−s, E, w, −k, b, −n, z, B, −p, d, −i, u,},

{B, −r, h, −c, m, −w, 0, w, −m, c, −h, r, −B, −B, r, −h, c, −m, w, 0,−w, m, −c, h, −r, B, B, −r, h, −c, m, −w,},

{C, −u, m, −e, d, −l, t, −B, −D, v, −n, f, −c, k, −s, A, E, −w, o, −g,b, −j, r, −z, −F, x, −p, h, −a, i, −q, y,},

{D, −x, r, −l, f, −a, g, −m, s, −y, E, C, −w, q, −k, e, −b, h, −n, t,−z, F, B, −v, p, −j, d, −c, i, −o, u, −A,},

{E, −A, w, −s, o, −k, g, −c, b, −f, j, −n, r, −v, z, −D, −F, B, −x, t,−p, l, −h, d, −a, e, −i, m, −q, u, −y, C,},

{F, −D, B, −z, x, −v, t, −r, p, −n, l, −j, h, −f, d, −b, a, −c, e, −g,i, −k, m, −o, q, −s, u, −w, y, −A, C, −E,},

}

An example value of the elements in 32-point DCT-8 core is: {a, b, c, d,e, f, g, h, i, j, k, 1, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B,C, D, E, F}={90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68,66, 63, 60, 56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4}

The techniques for encoding/decoding can be implemented by one or moreprocessors that may execute computer software having computer-readableinstructions that may be physically stored in one or morecomputer-readable media (e.g., a hard disk drive). For example, FIG. 7shows a computer system 700 suitable for implementing certainembodiments of the disclosed subject matter.

The computer software can be coded using any suitable machine code orcomputer language, that may be subject to assembly, compilation,linking, or like mechanisms to create code comprising instructions thatcan be executed directly, or through interpretation, micro-codeexecution, and the like, by computer central processing units (CPUs),Graphics Processing Units (GPUs), and the like.

The instructions can be executed on various types of computers orcomponents thereof, including, for example, personal computers, tabletcomputers, servers, smartphones, gaming devices, internet of thingsdevices, and the like.

The components shown in FIG. 7 for computer system 700 are exemplary innature and are not intended to suggest any limitation as to the scope ofuse or functionality of the computer software implementing embodimentsof the present disclosure. Neither should the configuration ofcomponents be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary embodiment of a computer system 700.

Computer system 700 may include certain human interface input devices.Such a human interface input device may be responsive to input by one ormore human users through, for example, tactile input (such as:keystrokes, swipes, data glove movements), audio input (such as: voice,clapping), visual input (such as: gestures), olfactory input (notdepicted). The human interface devices can also be used to capturecertain media not necessarily directly related to conscious input by ahuman, such as audio (such as: speech, music, ambient sound), images(such as: scanned images, photographic images obtain from a still imagecamera), video (such as two-dimensional video, three-dimensional videoincluding stereoscopic video).

Input human interface devices may include one or more of (only one ofeach depicted): keyboard 701, mouse 702, trackpad 703, touch screen 710,data-glove 704, joystick 705, microphone 706, scanner 707, camera 708.

Computer system 700 may also include certain human interface outputdevices. Such human interface output devices may be stimulating thesenses of one or more human users through, for example, tactile output,sound, light, and smell/taste. Such human interface output devices mayinclude tactile output devices (for example tactile feedback by thetouch-screen 710, data-glove 704, or joystick 705, but there can also betactile feedback devices that do not serve as input devices), audiooutput devices (such as: speakers 709, headphones (not depicted)),visual output devices (such as screens 710 to include CRT screens, LCDscreens, plasma screens, OLED screens, each with or without touch-screeninput capability, each with or without tactile feedback capability—someof which may be capable to output two dimensional visual output or morethan three dimensional output through means such as stereographicoutput; virtual-reality glasses (not depicted), holographic displays andsmoke tanks (not depicted)), and printers (not depicted).

Computer system 700 can also include human accessible storage devicesand their associated media such as optical media including CD/DVD ROM/RW720 with CD/DVD or the like media 721, thumb-drive 722, removable harddrive or solid state drive 723, legacy magnetic media such as tape andfloppy disc (not depicted), specialized ROM/ASIC/PLD based devices suchas security dongles (not depicted), and the like.

Those skilled in the art should also understand that term “computerreadable media” or “computer readable medium” as used in connection withthe presently disclosed subject matter does not encompass transmissionmedia, carrier waves, or other transitory signals.

Computer system 700 can also include interface to one or morecommunication networks 755. Networks 755 can for example be one or moreof wireless, wireline, or optical. Networks 755 can further be one ormore of local, wide-area, metropolitan, vehicular and industrial,real-time, or delay-tolerant, and so on. Examples of networks includelocal area networks such as Ethernet, wireless LANs, cellular networksto include GSM, 3G, 4G, 5G, LTE and the like, TV wireline or wirelesswide area digital networks to include cable TV, satellite TV, andterrestrial broadcast TV, vehicular and industrial to include CANBus,and so forth. Certain networks 755 commonly require external networkinterface adapters 754 that attached to certain general purpose dataports or peripheral buses (749) (such as, for example USB ports of thecomputer system 700; others are commonly integrated into the core of thecomputer system 700 by attachment to a system bus as described below(for example Ethernet interface into a PC computer system or cellularnetwork interface into a smartphone computer system). Using any of thesenetworks 755, computer system 700 can communicate with other entities.Such communication can be uni-directional, receive only (for example,broadcast TV), uni-directional send-only (for example CANbus to certainCANbus devices), or bi-directional, for example to other computersystems using local or wide area digital networks 755. Certain protocolsand protocol stacks can be used on each of those networks 755 andnetwork interfaces 754 as described above.

Aforementioned human interface devices, human-accessible storagedevices, and network interfaces can be attached to a core 740 of thecomputer system 700.

The core 740 can include one or more Central Processing Units (CPU) 741,Graphics Processing Units (GPU) 742, specialized programmable processingunits in the form of Field Programmable Gate Areas (FPGA) 743, hardwareaccelerators for certain tasks 744, and so forth. These devices, alongwith Read-only memory (ROM) 745, Random-access memory 746, internal massstorage such as internal non-user accessible hard drives, SSDs, and thelike 747, may be connected through a system bus 748. In some computersystems, the system bus 748 can be accessible in the form of one or morephysical plugs to enable extensions by additional CPUs, GPU, and thelike. The peripheral devices can be attached either directly to thecore's system bus 748, or through a peripheral bus 749. Architecturesfor a peripheral bus include PCI, USB, and the like.

CPUs 741, GPUs 742, FPGAs 743, and accelerators 744 can execute certaininstructions that, in combination, can make up the aforementionedcomputer code. That computer code can be stored in ROM 745 or RAM 746.Transitional data can also be stored in RAM 746, whereas permanent datacan be stored for example, in the internal mass storage 747. Faststorage and retrieve to any of the memory devices can be enabled throughthe use of cache memory, that can be closely associated with one or moreCPU 741, GPU 742, mass storage 747, ROM 745, RAM 746, and the like.

The computer readable media can have computer code thereon forperforming various computer-implemented operations. The media andcomputer code can be those specially designed and constructed for thepurposes of the present disclosure, or they can be of the kind wellknown and available to those having skill in the computer software arts.

As an example and not by way of limitation, the computer system havingarchitecture 700, and specifically the core 740 can providefunctionality as a result of processor(s) (including CPUs, GPUs, FPGA,accelerators, and the like) executing software embodied in one or moretangible, computer-readable media. Such computer-readable media can bemedia associated with user-accessible mass storage as introduced above,as well as certain storage of the core 740 that are of non-transitorynature, such as core-internal mass storage 747 or ROM 745. The softwareimplementing various embodiments of the present disclosure can be storedin such devices and executed by core 740. A computer-readable medium caninclude one or more memory devices or chips, according to particularneeds. The software can cause the core 740 and specifically theprocessors therein (including CPU, GPU, FPGA, and the like) to executeparticular processes or particular parts of particular processesdescribed herein, including defining data structures stored in RAM 746and modifying such data structures according to the processes defined bythe software. In addition or as an alternative, the computer system canprovide functionality as a result of logic hardwired or otherwiseembodied in a circuit (for example: accelerator 744), which can operatein place of or together with software to execute particular processes orparticular parts of particular processes described herein. Reference tosoftware can encompass logic, and vice versa, where appropriate.Reference to a computer-readable media can encompass a circuit (such asan integrated circuit (IC)) storing software for execution, a circuitembodying logic for execution, or both, where appropriate. The presentdisclosure encompasses any suitable combination of hardware andsoftware.

While this disclosure has described several exemplary embodiments, thereare alterations, permutations, and various substitute equivalents, whichfall within the scope of the disclosure. It will thus be appreciatedthat those skilled in the art will be able to devise numerous systemsand methods which, although not explicitly shown or described herein,embody the principles of the disclosure and are thus within the spiritand scope thereof.

The invention claimed is:
 1. A method of decoding or encoding, themethod comprising: receiving information regarding a target data blockfor encoding or decoding, the target data block for encoding or decodingbeing one of: a compressed video or image data block or an uncompressedvideo or image data block; determining whether to use a recursivetransform or a non-recursive transform for the encoding or decoding ofthe target data block; when a result of the determination is to use therecursive transform: determining a first portion of the recursivetransform based on a compound orthonormal transform (COT), deriving asecond portion of the recursive transform due to thesymmetry/anti-symmetry properties of the core of the recursive transformin view of the first portion of the COT, and causing or transmittinginformation that causes the target data block to be encoded or decodedusing the first portion and the second portion of the recursivetransform as the recursive transform.
 2. The method according to claim1, further comprising, when a result of the determination is to use thenon-recursive transform: deriving the non-recursive transform based onthe information regarding the non-recursive transform stored in the COT,and causing or transmitting information that causes the target datablock to be encoded or decoded using the derived non-recursivetransform.
 3. The method according to claim 2, wherein at least part ofthe non-recursive transform is embedded in at least part of one half ofthe larger recursive transform in the COT, and the size of thenon-recursive transform of the COT is M, where M=N/2, and N is the sizeof the recursive transform of the COT.
 4. The method according to claim2, wherein the non-recursive transform is: DST-7 or DCT-8.
 5. The methodaccording to claim 2, wherein the recursive transform is DCT-2 and thenon-recursive transforms is DST-7 or DCT-8.
 6. The method according toclaim 5, further comprising: creating a resulting basis that correspondsto the original odd bases of 2N-point DCT-2 by: (i) replacing the firsthalf of odd bases of 2N-point DCT-2 with N-point scaled DST-7 or DCT-8bases, and (ii) filling the second half of the odd bases of the 2N-pointDCT-2 with the flipped DST-7 or DCT-8 bases plus inverse signs.
 7. Themethod according to claim 1, wherein the recursive transform is one of:DCT-2, Hadamard transform, DCT-1, DST-1, DST-2, DCT-3 or DST-3.
 8. Themethod according to claim 7, wherein the recursive transform is DCT-2,and the method further comprises, when using the recursive DCT-2transform of the COT: (i) determining a left half of the DCT-2 transformmatrix based on the COT, and (ii) deriving a right half of the DCT-2transform matrix according to the symmetry/anti-symmetry property ofDCT-2 core in view of the left half of the DCT-2 transform matrix. 9.The method according to claim 1, further comprising: prior to using theCOT, generating the COT by embedding information regarding thenon-recursive transform into the recursive transform that has a largersize than the non-recursive transform, and storing the COT.
 10. Themethod according to claim 9, wherein the generating of the COT byembedding the information regarding the non-recursive transform into thelarger recursive transform and using the generated COT for the encodingor decoding is indicated in high-level syntax, which includes at leastone of: a Video Parameter Set (VPS), a Sequence Parameter Set (SPS), aPicture Parameter Set (PPS), a slice header, or a CTU header.
 11. Themethod according to claim 1, further comprising: generating the COT as anew orthonormal transform, by embedding an N-point DCT-8 or DST-7 into a2N-point DCT-2, which results in the new orthonormal transform which isstill an orthonormal transform due to the symmetric/anti-symmetriccharacteristics of DCT-2.
 12. The method according to claim 1, whereinDCT-4 is extracted from DCT-2, DCT-4 replaces DCT-8 in an AdaptiveMultiple Transform (AMT) for certain block sizes, the new COT is derivedby flipping DCT-4 transform bases left and right, and the new COTreplaces DST-7 in AMT for certain block sizes.
 13. The method accordingto claim 1, wherein the first portion of the recursive transform and thesecond portion of the recursive transform are each half portions of therecursive transform.
 14. An apparatus comprising: at least one memoryconfigured to store computer program code; and at least one processorconfigured to access the at least one memory and operate according tothe computer program code, the computer program code comprising: firstapplying code configured to cause the at least one processor to receiveinformation regarding a target data block for encoding or decoding, thetarget data block for encoding or decoding being one of: a compressedvideo or image data block or an uncompressed video or image data block;second applying code configured to cause the at least one processor todetermine, for the encoding or decoding of the target data block,whether to use a recursive transform or a non-recursive transform forthe encoding or decoding of the target data block; and third applyingcode configured to cause the at least one processor to when a result ofthe determination is to use the recursive transform: determine a firstportion of the recursive transform based on the COT, derive a secondportion of the recursive transform due to the symmetry/anti-symmetryproperties of the core of the recursive transform in view of the firstportion of the COT, and cause or transmit information that causes thetarget data block to be encoded or decoded using the first half portionand the second half portion of the recursive transform as the recursivetransform.
 15. The apparatus according to claim 14, further comprisingfourth applying code configured to cause the at least one processor towhen a result of the determination is to use the non-recursivetransform: deriving the non-recursive transform based on the informationregarding the non-recursive transform stored in the COT, and causing ortransmitting information that causes the target data block to be encodedor decoded using the derived non-recursive transform.
 16. The apparatusaccording to claim 15, wherein the recursive transform is one of: DCT-2,a Hadamard transform, DCT-1, DST-1, or DST-2, and the non-recursivetransform is: DST-7 or DCT-8.
 17. The apparatus according to claim 14,wherein at least part of the non-recursive transform is embedded in atleast part of one half of the larger recursive transform in the COT, andthe size of the non-recursive transform of the COT is M, where M=N/2,and N is the size of the recursive transform of the COT.
 18. Theapparatus according to claim 14, further comprising initial applyingcode configured to cause the at least one processor to generate the COTas a new orthonormal transform, by embedding an N-point non-recursivetransform into a 2N-point recursive transform, which results in the neworthonormal transform which is still an orthonormal transform due to thesymmetric/anti-symmetric characteristics of the recursive transform. 19.The apparatus according to claim 14, wherein the recursive transform isDCT-2, and the computer program code further comprises sixth applyingcode that is configured to cause the at least one processor to, whenusing the recursive DCT-2 transform of the COT: (i) determine the lefthalf of the DCT-2 transform matrix based on the COT, and (ii) derive theright half of the DCT-2 transform matrix according to thesymmetry/anti-symmetry property of DCT-2 core in view of the left halfof the DCT-2 transform matrix.
 20. A non-transitory computer-readablestorage medium storing instructions that, when executed by one or moreprocessors, causes the one or more processors to: receive informationregarding a target data block for encoding or decoding, the target datablock for encoding or decoding being one of: a compressed video or imagedata block or an uncompressed video or image data block; determine, forthe encoding or decoding of the target data block, whether to use therecursive transform or the non-recursive transform for the encoding ordecoding of the target data block; and when a result of thedetermination is to use the recursive transform: determine a firstportion of the recursive transform based on the COT, derive a secondportion of the recursive transform due to the symmetry/anti-symmetryproperties of the core of the recursive transform in view of the firstportion of the COT, and cause or transmit information that causes thetarget data block to be encoded or decoded using the first half portionand the second half portion of the recursive transform as the recursivetransform.